Title: Theory / Important / Math / Al gebra / Linear / done right;
Tags: fromMath mathToDef property hasSeenConjDef multipartite automaticImplies closed sides complex counterExample CorrectType implication fromAlgorithm hasNotNewEffectReturnsMeaning algorithmWithCardNameButNoFieldName2 equivalentHand mid1516974248815 toBinaryRelation EquivalentButNoIntermediate RelationToQuestion TypToQuestion addIntermediateCard defToAlgebra examplesSwitchExample1 list oldAlgoNewName setAdditionMultiplication toAlgebra toBinaryFunction toUnary

Name: \(n\)-tuple of elements of \(K\)
Definition: Some \((t_1,\dots,t_n)\) with \(t_i\in K\).
Page: 005
Chapter: 1
Section: A
Variables: \(K\) be a set
\(n\in\mathbb N\).
Index: 8
Kind: Def

Definition: The set of \(n\) tuples of elements of \(F\).
Page: 006
Chapter: 1
Section: A
Variables: \(F\) be a set,
\(n\in\mathbb N\).
Notation: \(F^n\)
Index: 10
Kind: Def

Definition: \((0,\dots,0)\).
Page: 007
Chapter: 1
Section: A
Variables: \(\mathbb F\) be a field
\(n\in\mathbb N\).
Notation: 0
Index: 14
Kind: Def

Page: 007
Section: A
Notation: +
Variables: \(\mathbb F\) be a commutative group
\(n\in\mathbb N\).
Chapter: 1
Name: Addition
Definition: \((x_1,\dots,x_n),(y_1,\dots,y_n)\mapsto(x_1+y_1,\dots,x_n+y_n)\)
Index: 13
Commutative: Commutative
Associative: Associative
Context: \(\mathbb F^n\)
Metric: No
Unit: \((0,\dots, 0)\)
Zero: None

Definition: \((\lambda x_1,\dots,\lambda x_n)\)
Page: 010
Chapter: 1
Section: A
Variables: \(\mathbb F\) a ring,
\(\lambda\in\mathbb F\)
\(x_1,\dots,x_n\in\mathbb F\).
Notation: \(\lambda(x_1,\dots,x_n)\)
Index: 17
Kind: Def

Name: A \(\mathbb K\)-vector space
Definition: \(1\cdot x=x\)
Page: 012/28-31
Chapter: 1/2
Section: B/1
Variables: \((\mathbb K,+,\times,0,1)\) a field
Index: 19
Kind: Def/
Definition2: \(a\cdot(x+y)=a\cdot x+a\cdot y\).
Example: \(\mathbb F^n\) and \(\mathbb F^{n\times m}\)
Conjdef: A commutative group \((V,+,0)\), plus an operation \(\cdot:K\times V\to V\) such that
Name2: A linear space
Example2: \(\mathbb C\) is a \(\mathbb R\)-vector space
Example3: For \(V,W\) two \(\mathbb K\)-vector spaces, the set of functions from \(V\) to \(W\), with pointwise addition and \(c\times f=s\mapsto c\times (f(s))\)
Example4: The space of polynomials over \(\mathbb F\)

Definition: The set of infinite sequences of elements of \(\mathbb F\)
Page: 013
Chapter: 1
Section: B
Variables: \(\mathbb F\) be a set
Notation: \(F^\infty\)
Index: 22
Kind: Ex

Definition: The vector space of functions from \(S\) to \(\mathbb F\).
Page: 014
Chapter: 1
Section: B
Variables: \(\mathbb F\) a field
\(S\) a set.
Notation: \(\mathbb F^S\)
Index: 23
Kind: Notation
Assuming: nothing

Definition: [$]\defFun{S}{\mathbb F}{x}{-x}[/$]
Page: 015
Chapter: 1
Section: B
Variables: \(\mathbb F\) a field.
\(S\) a set,
\(\vec x\in\mathbb F^S\)
Notation: \(-\vec x\)
Index: 27
Kind: Notation
Assuming: nothing

Name: A subspace of \(V\)
Definition: is also a \(K\)-vector space.
Page: 018
Chapter: 1
Section: C
Variables: \(\mathbb F\) a field
\(V\) a \(\mathbb F\)-vector space.
Index: 32, 34
Kind: Def
Definition2: \(0\in U\) and \(U\) closed by addition and by scalar multiplication.
Conjdef: A subset \(U\subseteq V\), which equivalently
Assuming: nothing

Name: The sum of \(U_1,\dots, U_n\)
Definition: The set of elements \(x_1+\dots +x_n\) for \(x_i\in U_i\).
Page: 020
Chapter: 1
Section: C
Variables: \(\mathbb F\) a field
\(V\) a \(\mathbb F\)-vector space.
\(n\in\mathbb N\) and
\(U_1,\dots,U_n\subseteq V\).
Index: 36
Kind: Def
Definition2: The smallest subspace containing the \(U_i\)'s
Assuming: nothing

Name: \(W\) as a direct sum
Definition: Two subspaces \(U,V\subseteq W\) such that \(U\cap V=\{0\}\) and \(U+V=W\)
Page: 021
Chapter: 1
Section: C
Variables: \(\mathbb F\) a field
\(W\) a \(\mathbb F\)-vector space
Notation: \(U\oplus V=W\)
Index: 40
Kind: Def
Assuming: nothing

Name: A linear combination of elements of \(W\)
Definition: A vector of the form \(a_1v_1+\dots+a_nv_n\) with the \(a_i\in\mathbb K\) and \(v_i\in W\)
Page: 028
Chapter: 2
Section: A
Variables: \(\mathbb F\) a field
\(V\) a \(\mathbb F\)-vector space
\(W\subseteq V\)
Index: 3
Kind: Def
Assuming: nothing

Name: The span of \(W\)
Definition: the set of linear combination of \(W\).
Page: 029
Chapter: 2
Section: A
Variables: the smallest subspace containing the \(U_i\)'s
Index: 5 7
Kind: Def
Definition2: the smallest subspace containing \(W\).
Assuming: Nothing

Name: A \(m\times n\) matrix
Definition: \[\left( \begin{array}{ccc}A_{1,1} & \dots & A_{1,n}\\\vdots & & \vdots \\A_{m,1} & \dots & A_{m,n} \end{array} \right)\]
Page: 030
Chapter: 3
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
Denoted by: \(\mathcal M\)

Name: A set of vectors which span \(V\)
Definition: A set \(S\subseteq V\) such that the span of \(S\) is \(V\).
Page: 030
Chapter: 2
Section: A
Variables: \(\mathbb F\) a field
\(V\) a vector space
Index: 8
Kind: Def
Assuming: nothing

Name: Finite dimensional vector space
Definition: A vector space such that there exists a finite list of vectors \(v_1,\dots,v_n\) which spans it.
Page: 030
Chapter: 2
Section: A
Variables: \(\mathbb F\) a field
Index: 10
Kind: Def
Assuming: nothing

Name: A polynomial with coefficient in \(\mathbb F\)
Definition: A term of the form \(x\mapsto\sum_{i=0}^na_ix^i\) with the \(a_i\) in \(\mathbb F\).
Page: 030
Chapter: 2
Section: A
Variables: \(\mathbb F\) a field
Index: 11 12
Kind: Def
Denoted by: \(p\)
Assuming: nothing

Name: The degree of \(p\)
Definition: \(-\infty\) if \(p=0\)
Page: 030
Chapter: 2
Section: A
Variables: \(\mathbb F\) a field
\(p\in\mathcal P(\mathbb F)\)
Index: 12
Kind: Def
Definition2: \(n\) if \(p=\sum a_nx^n\) and \(a_n\ne0\)
Assuming: nothing

Definition: The set of polynomials of degree at most \(n\).
Page: 031
Chapter: 2
Section: A
Variables: \(\mathbb F\) be a field
Notation: \(\mathcal P_n(F)\)
Index: 13
Kind: Def
Assuming: nothing

Name: Infinite-dimensional vector space
Definition: A vector space which is not finite-dimensional.
Page: 031
Chapter: 2
Section: A
Variables: \(\mathbb F\) a field
Index: 15
Kind: Def
Assuming: nothing

Name: A linearly independant set of vectors
Definition: Some vectors \(v_1,\dots,v_n\in V\) such that \(\sum_{i=1}^n a_iv_i=0\) implies the \(a_i\)'s are 0.
Page: 032
Chapter: 2
Section: A
Variables: \(\mathbb F\) a field
Index: 17
Kind: Def
Assuming: nothing

Name: A linear dependant list of vectors
Definition: A list of vectors which is not independant.
Page: 033
Chapter: 2
Section: A
Variables: \(\mathbb F\) a field
Index: 19, 21
Kind: Def, Lem
Definition2: \(v_j\in \text{span}(v_1,\dots,v_{j-1},v_{j+1},\dots,v_n)\) for some \(0<j\le n\)
Definition3: \(\text{span}(v_1,\dots,v_{j-1},v_j,v_{j+1},\dots,v_n)=\text{span}(v_1,\dots,v_{j-1},\dots,v_{j+1},\dots,v_n)\) for some \(0<j\le n\).

Name: Finite-dimensional space
Page: 27, 036, 46
Section: 4, A
Chapter: 2, 2, 2(Kenneth)
Index: 26, 2
Variables: \(\mathbb F\) a field
Kind: , Theo
Extra: Advanced linear algebra, brown, Theo 2, Ch 2.4 P 27
Done right, Ch 2.4, p 27. Ch 2.A, pag 36
Under: subspace
Under2: sum
Condition2: finite
Under3: product
Under4: dual
Under5: inverse of dual

Name: A basis of \(V\)
Definition: A list of independant vectors spaning \(V\).
Page: 039
Chapter: 2
Section: B
Variables: \(\mathbb F\) a field
\(V\) a \(\mathbb F\)-vector space
Index: 27, 29
Kind: Def
Definition2: A set \(A\subseteq V\) such that each vector can be uniquely written as \(\sum_{a\in A} av_a\) with \(v_a\in \mathbb F\), not zero a finite number of time.
Assuming: nothing

Smaller0: \(v_1,\dots, v_n\) contains a basis
Greater0: \(v_1,\dots,v_n\) is a spanning list
Page: 040
Chapter: 2
Section: B
Variables: \(\mathbb F\) a field
\(n\in\mathbb N\)
\(V\) a \(\mathbb F\)-vector space
\(v_1,\dots,v_n \in V\)
Index: 31
Smaller to greater: iff

Smaller0: \(v_1,\dots,v_n\) can be extended to a basis of \(V\)
Greater0: \(v_1,\dots,v_n\) is linearly independant
Page: 041
Chapter: 2
Section: B
Variables: \(\mathbb F\) a field
\(V\) a \(\mathbb F\)-vector space
\(n\in\mathbb N\)
\(v_1,\dots,v_n \in V\)
Index: 33
Smaller to greater: iff
Assuming: \(V\) finite-dimensional

Question: \(V\) admits
Answer: a basis.
Page: 041
Chapter: 2
Section: B
Variables: \(\mathbb F\) a vector space
\(V\) a \(\mathbb F\)-vector space
Index: 32
Assuming: \(V\) finite-dimensional

Question: There exists a subspace \(W\) of \(V\) such that
Answer: \(V=U\oplus W\).
Page: 042
Chapter: 2
Section: B
Variables: \(\mathbb F\) a field
\(V\) a \(\mathbb F\)-vector space and
\(U\) a subspace
Index: 34
Assuming: \(V\) finite dimensional

Question: All basis of \(V\) have the property
Answer: to have the same length.
Page: 044
Chapter: 2
Section: C
Variables: \(\mathbb F\) a field
\(V\) a \(\mathbb F\)-vector space
Index: 35
Assuming: \(V\) finite dimensional

Name: The dimension of \(V\)
Definition: The length of any basis of the vector space.
Page: 044
Chapter: 2
Section: C
Variables: \(\mathbb F\) a field
\(V\) a \(\mathbb F\)-vector space
Index: 36
Kind: Def
Abbreviation: \(\text{dim }V\)
Assuming: \(V\) finite dimensional

Smaller0: \(\text{dim }U\)
Greater0: \(\text{dim }V\)
Page: 045
Chapter: 2
Section: C
Variables: \(\mathbb F\) a field
\(V\) a \(\mathbb F\)-vector space
\(U\) a subspace of \(V\)
Index: 38
Smaller to greater: \(\le\)
Assuming: \(V\) finite dimensional
Greater to smaller: \(\ge\)
Hide sides: yes

Smaller0: the sequence of \(v_i\)'s is linearly independant and \(n=\text{dim} V\)
Greater0: a basis
Page: 045
Chapter: 2
Section: C
Variables: \(\mathbb F\) a field
\(n\in\mathbb N\)
\(V\) a \(\mathbb F\)-vector space
\(v_1,\dots,v_n \in V\)
Index: 39 .43
Hide middle: yes
Prefix Greater: \((v_i)_{0<i\le n}\) is
Assuming: \(V\) finite dimensional
Smaller1: the sequence of \(v_i\)'s span \(V\) and \(n=\text{dim} V\)
Connect Smaller0: ,
or

Smaller0: \(\text{Dim } U_1+\text{Dim } U_2-\text{Dim }(U_1\cap U_2)\)
Greater0: \(\text{Dim }(U_1+U_2)\)
Page: 047
Chapter: 2
Section: C
Variables: \(\mathbb F\) a field
\(U_1, U_2\), \(\mathbb F\)-vector space
Index: 43
Smaller to greater: equals
Assuming: \(U_i\) finite dimensional

Name: Additive function
Definition: \(f:V\to W\) such that \(f(x+y)=f(x)+f(y)\).
Page: 052
Chapter: 3
Section: A
Variables: \(\mathbb F\) a field
\(V,W\) a \(\mathbb F\)-vector space
Index: 2
Kind: Def
Assuming: nothing

Name: An homogene function from \(V\) to \(W\)
Definition: \(f:V\to W\) such that for \(\lambda\) a scalar and \(v\) a vector, \(f(\lambda v)=\lambda f(v)\).
Page: 052
Chapter: 3
Section: A
Variables: \(\mathbb F\) a field
\(V,W\) a \(\mathbb F\)-vector space
Index: 2
Kind: Def
Assuming: nothing

Name: Set of linear map from \(V\) to \(W\)
Page: 052
Chapter: 3
Section: A
Variables: \(\mathbb F\) a field
\(V,W\) a \(\mathbb F\)-vector space
Notation: \(\mathcal L(V,W)\)
Index: 3 7
Kind: Notation
Typ: A \(\mathbb F\)-vector space
Assuming: nothing

Name: A linear map
Definition: An additive homogene map.
Page: 052
Chapter: 3
Section: A
Variables: \(\mathbb F\) a field
\(V\) a \(\mathbb F\)-vector space
Index: 2, 4
Kind: Def, Ex
Example: Zero map
Example2: identity map
Assuming: Nothing

Name: Linear map over \(F[x]\)
Page: 053
Chapter: 3
Section: A
Variables: \(\mathbb F\) a field
Index: 4
Kind: Example
Example: Differentiation
Example2: integration
Example3: multiplication by a polynomial

Question: The number of linear map from \(U\) to \(V\) equal to \(p\) on \(B\) is
Answer: exactly one.
Page: 054
Chapter: 3
Section: A
Variables: \(\mathbb F\) be a field
\(U,V\), \(\mathbb F\)-vector spaces
\(B\) a basis of \(U\)
\(p:B\to U\)
Index: 5

Definition: \(v\mapsto f(v)+g(v)\).
Page: 055
Chapter: 3
Section: A
Variables: \(\mathbb F\) a field
\(V,W\) two \(\mathbb F\)-vector space
\(f,g\in\mathcal L(V,W)\).
Notation: \(f+g\)
Index: 6
Kind: Def
Assuming: nothing

Definition: \(x\mapsto\lambda(f(x))\).
Page: 055
Chapter: 3
Section: A
Variables: \(\mathbb F\) a field
\(V,W\) two \(\mathbb F\)-vector spaces
\(f\in\mathcal L(V,W)\)
\(\lambda\in F\).
Notation: \(\lambda f\)
Index: 6
Kind: Def
Assuming: nothing

Definition: \(x\mapsto f(g(x))\).
Page: 055
Chapter: 3
Section: A
Variables: \(\mathbb F\) a field
\(V,W\) two \(\mathbb F\)-vector space
\(f\in\mathcal L(U,V)\)
\(g\in\mathcal L(V,W)\).
Notation: \(fg\)
Index: 8
Kind: Def
Assuming: Nothing
DefType: \(\mathcal L(U,W)\)

Definition: [$]\defFun{V}{V}{x}{x}[/$]
Page: 056
Section: A
Chapter: 3
Variables: \(\mathbb F\) a field
\(V\) two \(\mathbb F\)-vector spaces
Notation: \(I\)
Index: 9
Context: Vector space
Assuming: nothing

Question: Multiplication of linear maps is
Answer: associative
Page: 056
Chapter: 3
Section: A
Answer2: admits an identity
Index: 9
Answer3: distributive over addition.

Question: \(f(0)=\)
Answer: \(0\).
Page: 057
Chapter: 3
Section: A
Variables: \(\mathbb F\) a field
\(V,W\) two \(\mathbb F\)-vector spaces
\(f\in\mathcal L(V,W)\)
Index: 11

Name: Null space of \(T\)
Definition: The set of vectors mapped to 0. \(\{v\in V: Tv=0\}\).
Page: 059, 60
Chapter: 3
Section: B
Variables: \(\mathbb F\) a field
\(V,W\) two \(\mathbb F\)-vector space
\(T\in\mathcal L(V,W)\)
Index: 12, 14
Kind: Def
Name2: kernel of \(T\)
Abbreviation: null \(T\)
Typ: a subspace of \(V\).
Assuming: nothing

Name: The range of \(T\)
Definition: \(\{T x:x\in V\}\).
Page: 061, 62
Chapter: 3
Section: B
Variables: \(\mathbb F\) a field
\(V,W\) two \(\mathbb F\)-vector spaces,
\(T\in\mathcal L(V,W)\).
Index: 17, 19
Kind: Definition
Abbreviation: \(\text{range } T\)
Typ: A subspace of \(W\)
Assuming: nothing

Smaller0: \(\text{null }T=\{0\}\)
Greater0: \(T\) is injective (one-to-one)
Page: 61
Chapter: 3
Section: B
Variables: \(\mathbb F\) vector spaces
\(V,W\) two \(\mathbb F\)-vector space
\(T\in\mathcal L(V,W)\)
Index: 16
Smaller to greater: iff

Smaller0: \(\text{dim }V\)
Greater0: \(\text{dim ker}T+\text{dim im}T\)
Page: 063
Chapter: 3
Section: B
Variables: \(\mathbb F\) a field
\(V,W\) two \(\mathbb F\)-vector space
\(T\in\mathcal L(V,W)\)
Index: 22
Smaller to greater: =
Assuming: \(V\) finite dimensional
Name: Fundamental theorem of linear maps

Smaller0: \(\text{dim } W<\text{dim }V\)
Greater0: not injective
Page: 064
Chapter: 3
Section: B
Variables: \(\mathbb F\) a field
\(V,W\) two \(\mathbb F\)-vector space
\(T\in\mathcal L(V,W)\)
Index: 23
Prefix Greater: \(T\) is
Assuming: \(V,W\) finite dimensional

Smaller0: \(\text{dim }V<\text{dim }W\)
Greater0: \(T\) is not surjective
Page: 064
Chapter: 3
Section: B
Variables: \(\mathbb F\) a field
\(V,W\) two \(\mathbb F\)-vector space
\(T\in\mathcal L(V,W)\)
Index: 24
Assuming: \(V,W\) finite-dimensional

Name: An homogeneous system of linear equations
Definition: A list of linear equations whose constant term is 0.
Page: 065
Chapter: 3
Section: B
Index: 25
Kind: Ex

Name: An inhomogeneous system of linear equations
Definition: A list of linear equations which is not homogeneous.
Page: 065
Chapter: 3
Section: B
Index: 25
Kind: Ex

Smaller0: \(H\) have more variables than equations
Greater0: \(H\) has nonzero solutions
Page: 065
Chapter: 3
Section: B
Variables: \(H\) be a homogeneous system of linear equation
Index: 26
Counterexample: \(x+y=0\) three times

Smaller0: \(H\) has more equations than variables
Greater0: \(H\) has no solution for some choice of the constant terms
Page: 066
Chapter: 3
Section: C
Variables: \(H\) be a inhomogeneous system of linear equation
Index: 29
Counterexample: \(x=4\), \(x=7\) and \(y+z+t=0\)

Name: The matrix of \(T\) with respect to these basis
Definition: \(A=\mathcal M(T)\) defined by \(T v_k=\sum _{i=1}^mA_{i,k}w_i\).
Page: 070
Chapter: 3
Section: C
Variables: \(\mathbb F\) a field
\(V,W\) two \(\mathbb F\)-vector spaces with basis \(v_1,\dots,v_n\) and \(w_1,\dots,w_m\)
\(T\in\mathcal L(V,W)\).
Index: 32
Kind: Definition
Assuming: nothing

Definition: \[ \left( \begin{array}{ccc} A_{1,1}+C_{1,1} & \dots & A_{1,n}+C_{1,n} \\ \vdots & & \vdots \\ A_{m,1}+C_{m,1} & \dots & A_{m,n}+C_{m,n} \end{array} \right)\]
Page: 072
Chapter: 3
Section: C
Variables: \(\mathbb F\) a field,
Let \(m,n\in\mathbb N\),
Let \(A,C\in\mathbb F^m\)
Notation: \[ \left( \begin{array}{ccc} A_{1,1} & \dots & A_{1,n} \\ \vdots & & \vdots \\ A_{m,1} & \dots & A_{m,n} \end{array} \right) + \left( \begin{array}{ccc} C_{1,1} & \dots & C_{1,n} \\ \vdots & & \vdots \\ C_{m,1} & \dots & C_{m,n} \end{array} \right)\]
Index: 35
Kind: Definition

Definition: \(\mathcal M(S)+\mathcal M(T)\).
Page: 073
Chapter: 3
Section: C
Variables: \(\mathbb F\) a field
\(V,W\) two \(\mathbb F\)-vector spaces with basis \(v_1,\dots,v_n\) and \(w_1,\dots,w_m\), and
\(S,T\in\mathcal L(V,W)\).
Notation: \(\mathcal M(S+T)\)
Index: 36
Assuming: nothing

Question: The dimension of \(\mathbb F^{m,n}\) is
Answer: mn.
Page: 073
Chapter: 3
Section: C
Variables: \(\mathbb F\) a field
Index: 40

Smaller0: \[ \lambda \left( \begin{array}{ccc} A_{1,1} & \dots & A_{1,n} \\ \vdots & & \vdots \\ A_{m,1} & \dots & A_{m,n} \end{array} \right)\]
Greater0: \[\left( \begin{array}{ccc} \lambda A_{1,1} & \dots &\lambda A_{1,n} \\ \vdots & & \vdots \\ \lambda A_{m,1} & \dots &\lambda A_{m,n} \end{array} \right)\]
Page: 073
Chapter: 3
Section: C
Index: 37
Kind: Definition
Smaller to greater: =

Smaller0: \(\mathcal M(\lambda T)\)
Greater0: \(\lambda\mathcal M(T)\)
Page: 073
Chapter: 3
Section: C
Variables: \(\mathbb F\) a field
\(V,W\) two \(\mathbb F\)-vector spaces with basis \(v_1,\dots,v_n\) and \(w_1,\dots,w_m\), and
\(T\in\mathcal L(V,W)\),
\(\lambda\in F\)
Index: 38
Smaller to greater: =
Assuming: nothing

Definition: The set of \(m\)-by-\(n\) matrix with entries in \(\mathbb F\).
Page: 073
Chapter: 3
Section: C
Variables: \(\mathbb F\) a field
\(m,n\in\mathbb N\)
Notation: \(\mathbb F^{m,n}\)
Index: 39
Kind: Notation
Assuming: nothing

Smaller0: \(\mathcal M(ST)\)
Greater0: \(\mathcal M(S)\mathcal M(T)\)
Page: 075
Chapter: 3
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\),
\(V,W\) two \(\mathbb F\)-vector spaces with basis \(u_1,\dots,u_n\), \(v_1,\dots,v_n\) and \(w_1,\dots,w_m\), and
\(T\in\mathcal L(U,V)\), \(S\in\mathcal L(V,W)\).
Index: 43
Smaller to greater: =
Assuming: nothing

Definition: The matrix \(M\in\mathbb F^{m,p}\) equal to
\[M_{j,k}=\sum_{r=1}^nA_{j,r}C_{r,k}.\]
Page: 075
Chapter: 3
Section: C
Variables: \(\mathbb F\) a field,
Let \(A\in\mathbb F^{m,n}\) and
\(C\in\mathbb F^{n,p}\).
Notation: \(AC\)
Index: 41
Kind: Definition

Definition: The \(i\)-th row of \(A\).
Page: 076
Chapter: 77
Section: C
Variables: \(\mathbb F\) a field,
Let \(m,n\in\mathbb N\).
Let \(A\in\mathbb F^{m,n}\).
Let \(i\in\mathbb N\).
Notation: \(A_{i,.}\)
Index: 44
Kind: Notation
Assuming: nothing

Definition: The \(i\)-th column of \(A\).
Page: 076
Chapter: 3
Section: C
Variables: \(\mathbb F\) a field,
\(m,n\in\mathbb N\).
\(A\in\mathbb F^{m,n}\).
\(i\in\mathbb N\).
Notation: \(A_{.,i}\)
Index: 44
Kind: Notation
Assuming: nothing

Smaller0: \((AC)_{j,k}\)
Greater0: \(A_{j,.}C_{.,k}\)
Page: 076
Chapter: 3
Section: C
Variables: \(\mathbb F\) a field,
Let \(m,n,p\in\mathbb N\).
Let \(A\in\mathbb F^{m,n}\) and
\(C\in\mathbb F^{n,p}\).
Let \(0<j\le m,0<k\le p\).
Index: 47
Smaller to greater: =
Assuming: nothing

Smaller0: \((AC)_{.,k}\)
Greater0: \(A(C_{.,k})\)
Page: 077
Chapter: 3
Section: C
Variables: \(\mathbb F\) a field,
Let \(m,n,p\in\mathbb N\).
Let \(A\in\mathbb F^{m,n}\) and
\(C\in\mathbb F^{n,p}\).
Let \(0<k\le p\).
Index: 49
Smaller to greater: =
Hide middle: y
Assuming: nothing

Smaller0: \(Ac\)
Greater0: \(\sum_{i=1}^nc_i A_{.,i}\)
Page: 77
Chapter: 3
Section: C
Variables: \(m,n\in\mathbb N\),
\(\mathbb F\) a field,
\(A\in\mathbb F^{m\times n}\),
\(c=\left(\begin{array}{l}c_1\\ \vdots\\c_n\end{array}\right)\)
Smaller to greater: =
Hide middle: yes

Name: An invertible linear map
Definition: admits an inverse
Page: 80, 81, 82, 87/ 23
Chapter: 3/ 1 kunze
Section: D/6
Variables: \(m,n\in\mathbb N\),
\(\mathbb F\) a field,
\(V,W\) be \(\mathbb F\) vector spaces
Index: 53, 56, 58, 69/ 12, 13
Kind: / theo,
Definition2: is bijective
Conjdef: A linear map \(A\) which, equivalently
Definition3: is injective/surjective (assuming finite dimensional)
Name2: An isomorphism
Definition4: row equivalent to the \(n\times n\) identity matrix (assuming finite dimensional)
Definition5: product of elementary matrices (assuming finite dimensional)
Definition6: \(AX=0\) has only the trivial solution (assuming finite dimensional)
Counterexample: multiplication by \(x^2\) in \(\mathbb F[x]\)
Definition7: \(AX=Y\) has a solution \(X\) for each \(Y\in\mathbb F^N\) (assuming finite dimensional)
Counterexample2: backward shift in \(\mathbb F^\infty\)

Name: An inverse of \(T\)
Definition: A map \(S\in\mathcal L(W,V)\) such that \(ST\) and \(TS\) are identity maps.
Page: 80
Chapter: 3
Section: D
Variables: \(\mathbb F\) a field,
\(V,W\) be \(\mathbb F\) vector spaces
\(T\in\mathcal L(V,W)\)
Index: 53, 54
Cardinal: At most one

Page: 82,
Definition: that there is an isomorphism from one onto the other
Chapter: 3
Section: D
Variables: \(\mathbb F\) a field,
Name: Isomorphic vector spaces
Index: 58, 59
Definition2: having the same dimension (assuming it is finite)
Equivalence: yes
Set: Vectors spaces

Smaller0: \(\mathcal L(V,W)\)
Greater0: \(\mathbb F^{\text{dim }V\times\text{dim }W}\)
Page: 83
Chapter: 3
Section: D
Variables: \(\mathbb F\) a field,
\(V,W\) be \(\mathbb F\) vector spaces
Smaller to greater: is isomorphic to
Assuming: finite dimensional

Smaller0: \(\text{dim }\mathcal L(V,W)\)
Greater0: \(\text{dim }V\times\text{dim }W\)
Page: 83
Chapter: 3
Section: D
Variables: \(\mathbb F\) a field,
\(V,W\) be \(\mathbb F\) vector spaces
Index: 61
Smaller to greater: equals
Assuming: finite dimensional

Name: The matrix of \(v\)
Definition: \(\left(\begin{array}{l}c_1\\ \vdots\\c_n\end{array}\right)\) such that \(v=\sum c_iv_i\)
Page: 84
Chapter: 3
Section: D
Variables: \(\mathbb F\) a field,
\(V\) be \(\mathbb F\) vector spaces with basis \(v_1,\dots,v_n\)
\(v\in V\)
Notation: \(\mathcal M(v)\)
Index: 62

Smaller0: \(\mathcal M(Tv)\)
Greater0: \(\mathcal M(T)\mathcal M(v)\)
Page: 85
Chapter: 3
Section: D
Variables: \(\mathbb F\) a field,
\(V,W\) be \(\mathbb F\) vector spaces with basis \(v_1,\dots,v_n\) and \(w_1,\dots,w_m\)
\(v\in V\)
\(T\in\mathcal L(V,W)\)
Index: 65
Smaller to greater: =

Name: An operator
Definition: A linear map from a vector space to itself
Page: 86
Chapter: 3
Section: D
Index: 67

Definition: \(\mathcal L(V,V)\)
Page: 86
Chapter: 3
Section: D
Variables: \(\mathbb F\) a field,
\(V\) be \(\mathbb F\) vector spaces with basis \(v_1,\dots,v_n\)
Notation: \(\mathcal L(V)\)
Index: 67

Smaller0: injective
Greater0: the \(V_i\)'s are in direct sum
Page: 93
Chapter: 3
Section: E
Variables: \(\mathbb F\) a field,
\(\mathcal I\) an index set
\(W\) a \(\mathbb F\)-vector space
\((V_i)_{i\in\mathcal I}\subseteq W\) be subspaces
Index: 77, 78
Smaller to greater: iff
Prefix Smaller: [$]\defFun{\prod V_i}{W}{(v_i)}{\sum v_i}[/$] is
Intermediate: \(\text{dim }\sum V_i=\sum\text{dim }V_i\)

Definition: the set of \(v+w\) with \(v\in V\)
Page: 94
Chapter: 3
Section: E
Variables: \(\mathbb F\) a field,
\(\mathcal I\) an index set
\(W\) a \(\mathbb F\)-vector space
\(V\subseteq W\) a subspace
\(w\in W\)
Notation: \(w+V\)
Index: 79

Name: An affine subset of \(W\)
Definition: A set \(w+V\) for some \(w\in W\), V a subspace
Page: 94
Chapter: 3
Section: E
Variables: \(\mathbb F\) a field,
\(\mathcal I\) an index set
\(W\) a \(\mathbb F\)-vector space
Index: 81

Page: 94
Definition: The relations between affine subsets of the form \(w+V\) and \(w'+V\) for some \(w,w'\in W\), \(V\subseteq W\) subspaces
Chapter: 3
Section: E
Variables: \(\mathbb F\) a field,
\(\mathcal I\) an index set
\(W\) a \(\mathbb F\)-vector space
Name: Parallel affine subset of \(W\)
Index: 81

Name: The quotient space of \(V\) by \(U\)
Definition: elements are \(v+U\) for \(v\in V\)
Page: 95
Chapter: 3
Section: E
Variables: \(\mathbb F\) a field,
\(\mathcal I\) an index set
\(V\subseteq W\) a \(\mathbb F\)-vector space
Notation: \(V/U\)
Index: 83
Definition2: \(((u+V),(w+U))\mapsto (u+w)+U\)
Conjdef: The \(\mathbb F\) vector space
Definition3: \(\lambda,(u+U)\mapsto (\lambda u)+U\)
Property: Dimension \(\text{dim }U-\text{dim }V\)

Smaller0: \(v-w\in U\)
Greater0: \(v+U=w+U\)
Page: 95-97
Chapter: 3
Section: E
Variables: \(\mathbb F\) a field,
\(\mathcal I\) an index set
\(V\subseteq W\) a \(\mathbb F\)-vector space,
\(w,w'\in W\)
Index: 85, 87, 89
Smaller to greater: iff
Intermediate: \((v+U)\) intersects \((w+U)\)

Name: The quotient map
Definition: The linear map [$]\defFun V{V/U}{v}{v+U}[/$].
Page: 097
Chapter: 3
Section: E
Variables: \(\mathbb F\) a field,
\(V\) a \(\mathbb F\)-vector spaces,
\(U\) a subspace of \(V\)
Notation: \(\pi\)
Index: 88
Kind: Definition
Assuming: nothing

Question: For \(T\in\mathcal L(V,W)\), the range of \(\tilde T\) is
Answer: the range of \(T\).
Page: 098
Chapter: 3
Section: E
Variables: \(\mathbb F\) a field
\(V,W\) \(F\)-vector space
\(T\in\mathcal L(V,W)\)
Index: 91 c
Assuming: Nothing

Question: For \(T\in\mathcal L(V,W)\), \(\tilde T\) is
Answer: injective.
Page: 098
Chapter: 3
Section: E
Variables: \(\mathbb F\) a field
\(V,W\) two \(F\)-vector space and
\(T\in\mathcal L(V,W)\).
Index: 91 b
Assuming: Nothing

Definition: [$]\defFun {V/\text{null } T}W{u+\text{null }T}{T(u)}[/$].
Page: 098
Chapter: 3
Variables: \(\mathbb F\) a field,
\(V\) a \(\mathbb F\)-vector spaces,
\(T\in\mathcal L(V,W)\)
Notation: \(\tilde{T}\)
Index: 91
Kind: E
Assuming: nothing

Name: A linear functional
Definition: A linear map from \(V\) to \(\mathbb F\)
Page: 101
Chapter: 3
Variables: \(\mathbb F\) a field
\(V\) a \(\mathbb F\)-vector spaces
Index: 92
Kind: F
Definition2: An element of \(\mathcal L(V,\mathbb F)\)

Name: Dual space of \(V\)
Definition: \(\mathcal L(V,\mathbb F)\).
Page: 101
Chapter: 3
Variables: \(\mathbb F\) a field,
\(V\) a \(\mathbb F\)-vector spaces
Notation: \(V'\)
Index: 94
Kind: Definition
Assuming: nothing

Question: The dimension of \(V'\) is
Answer: the dimension of \(V\).
Page: 101
Chapter: 3
Section: E
Variables: \(\mathbb F\) a field
\(V\) a \(\mathbb F\)-vector spaces
Index: 95
Assuming: \(V\) finite dimensional

Name: The dual basis of \(v_1,\dots,v_n\)
Definition: The list \(\phi_1,\dots,\phi_n\) of elements of \(V'\), where \(\phi_j\) is the linear functional on \(V\) such that
\( \phi_j(v_k)= \left\{ \begin{array}{ll} 1& \text{ if }k=j\\ 0& \text{ if }k\neq j \end{array} \right.\)
Page: 102
Chapter: 3
Section: E
Variables: \(\mathbb F\) a field,
\(V\) a \(\mathbb F\)-vector spaces
\(v_1,\dots,v_n\in V\)
Index: 96 98
Typ: A basis of \(V'\)
Assuming: \(v_1,\dots,v_n\) a basis of \(V\)

Name: The dual map of \(T\)
Definition: \(\phi\mapsto\phi\circ T\).
Page: 103
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field
\(V,W\) \(F\)-vector space
\(T\in\mathcal L(V,W)\)
Notation: \(T'\)
Index: 99
Kind: Definition
Assuming: nothing
DefType: \(\mathcal L(W',V')\)
Applied to: f

Smaller0: \((ST)'\)
Greater0: \(T'S'\)
Page: 104
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field,
\(U,V,W\) a \(\mathbb F\)-vector spaces,
\(S\in\mathcal L(U,V),T\in\mathcal L(V,W)\)
Index: 101
Smaller to greater: =
Hide middle: yes
Assuming: nothing

Name: The annihilator of \(U\)
Definition: \[\{\phi\in V': \phi(u)=0 \text{ for all } u\in U\}.\]
Page: 104, 105
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field,
\(V\) a \(\mathbb F\)-vector spaces,
\(U\) a subset of \(V\)
Notation: \(U^0\)
Index: 102, 105
Kind: Definition
Typ: A subspace of \(V'\)
Assuming: Nothing

Smaller0: \((S+T)'\)
Greater0: \(S'+T'\)
Page: 104
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field
\(V,W\) \(F\)-vector space
\(T,U\in\mathcal L(V,W)\)
Index: 101
Smaller to greater: =
Hide middle: yes
Assuming: Nothing

Smaller0: \((\lambda T)'\)
Greater0: \(\lambda T'\)
Page: 104
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field
\(V,W\) a \(\mathbb F\)-vector spaces,
\(T\in\mathcal L(V,W)\)
Index: 101
Smaller to greater: =
Hide middle: yes

Question: \(\text{dim }U^0=\)
Answer: \(\text{dim }V-\text{dim }U\).
Page: 106
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field,
\(V\) two \(F\)-vector space and
\(U\) a subspace of \(V\)
Assuming: \(V\) finite dimensional
Extra: répétition

Smaller0: \(\text{null }T'\)
Greater0: \((\text{range }T)^0\)
Page: 106
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field,
\(V,W\) a \(\mathbb F\)-vector spaces,
\(T\in\mathcal L(V,W)\)
Index: 107
Smaller to greater: =
Assuming: \(V,W\) finite-dimensional

Question: \(\left(\text{dim null }T'\right)=\)
Answer: \(\text{dim null }T+\text{dim }W-\text{dim }V\).
Page: 106
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field
\(V,W\) \(F\)-vector space
\(T\in\mathcal L(V,W)\)
Index: 107
Assuming: \(V,W\) finite dimensional

Smaller0: \(T\) surjective
Greater0: \(T'\) being injective
Page: 107
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field
\(V,W\) \(F\)-vector space
\(T\in\mathcal L(V,W)\)
Index: 108
Smaller to greater: iff
Assuming: \(V,W\) finite dimensional

Question: \(\text{dim range }T'=\)
Answer: \(\text{dim range }T\).
Page: 107
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field
\(V,W\) \(F\)-vector space
\(T\in\mathcal L(V,W)\)
Index: 109
Assuming: \(V,W\) finite dimensional

Smaller0: \(\text{range }T'\)
Greater0: \((\text{null }T)^0\)
Page: 107
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field
\(V,W\) \(F\)-vector space
\(T\in\mathcal L(V,W)\)
Index: 109
Smaller to greater: =
Hide middle: yes
Assuming: \(V,W\) finite dimensional

Smaller0: surjective
Greater0: \(T\) is injective
Page: 108
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field
\(V,W\) \(F\)-vector space
\(T\in\mathcal L(V,W)\)
Index: 110
Smaller to greater: iff
Assuming: \(V,W\) finite dimensional
Prefix Smaller: \(T'\) is

Name: The transpose of \(A\)
Definition: The \(n\times m\) matrix whose \((k,j)\)-th component is \(A_{j,k}\).
Page: 109
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field,
\(A\in F^{m\times n}\)
Notation: \(A^t\)
Index: 111
Kind: Definition
Assuming: nothing

Smaller0: \((AC)^t\)
Greater0: \(C^tA^t\)
Page: 109
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field,
\(A\in F^{m\times n}\),
\(C\in\mathbb F ^{n\times p}\)
Index: 113
Smaller to greater: =
Hide middle: yes

Smaller0: \(\mathcal M(T')\)
Greater0: \((\mathcal M(T))^t\)
Page: 110
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field,
\(U,V\) \(\mathbb F\)-vector spaces with basis \(u_1,\dots,u_n\) and \(v_1,\dots,v_m\),
\(T\in\mathcal L(U,V)\)
Index: 114
Smaller to greater: =
Hide middle: yes

Smaller0: the row rank
Greater0: the column ranks
Page: 111
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field,
\(U,V\) two \(F\)-vector space
\(T\in\mathcal L(U,V)\).
Index: 117, 118
Smaller to greater: equals
Assuming: \(U,V\) finite dimensional
Smallest: \(\text{dim range }T\)

Name: The row rank of a map \(F\)
Definition: the dimension of the span of the rows of \(A\) in \(\mathbb F^{1,n}\).
Page: 111
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field
\(U,V\) \(\mathbb F\)-vector spaces with basis \(u_1,\dots,u_n\) and \(v_1,\dots,v_m\)
\(T\in\mathcal L(U,V)\)
Index: 115
Kind: Definition
Assuming: nothing

Name: The column rank of a map \(F\)
Definition: the dimension of the span of the columns of \(A\) in \(\mathbb F^{m,1}\).
Page: 111
Chapter: 3
Section: F
Variables: \(\mathbb F\) a field
\(U,V\) \(\mathbb F\)-vector spaces with basis \(u_1,\dots,u_n\) and \(v_1,\dots,v_m\)
\(T\in\mathcal L(U,V)\)
Index: 115
Kind: Definition
Assuming: nothing

Smaller0: \(z+\overline z\)
Greater0: \(2\text{Re} z\)
Page: 119
Chapter: 4
Section: A
Variables: \(z\in\mathbb C\)
Index: 5
Smaller to greater: =

Smaller0: \(z-\overline z \)
Greater0: \(2(\text{Im }z)i \)
Page: 119
Chapter: 4
Section: A
Variables: \(z\in\mathbb C\).
Index: 5
Smaller to greater: =

Smaller0: \(z\overline z\)
Greater0: \(|z|^2\)
Page: 119
Chapter: 4
Section: A
Variables: \(z\in\mathbb C\)
Index: 5
Smaller to greater: =

Question: \(\overline{\overline{z} }=\)
Answer: \(z\)
Page: 119
Chapter: 4
Section: A
Variables: \(z\in\mathbb C\)
Index: 5

Smaller0: \(\overline{w+z}\)
Greater0: \(\overline{w}+\overline z\)
Page: 119
Chapter: 4
Section: A
Variables: \(w,z\in\mathbb C\).
Index: 5
Smaller to greater: =

Smaller0: \(\overline{wz}\)
Greater0: \(\overline{w}\ \overline z\)
Page: 119
Chapter: 4
Section: A
Variables: \(w,z\in\mathbb C\).
Index: 5
Smaller to greater: =

Smaller0: \(|\overline z|\)
Greater0: \(|z|\)
Page: 119
Chapter: 4
Section: A
Variables: \(z\in\mathbb C\)
Index: 5
Smaller to greater: =

Smaller0: \(|wz|\)
Greater0: \(|w||z|\)
Page: 119
Chapter: 4
Section: A
Variables: \(w,z\in\mathbb C\).
Index: 5
Smaller to greater: =

Smaller0: \(|\text{Re }z|\)
Greater0: \(|z|\)
Page: 119
Chapter: 4
Section: A
Variables: \(z\in\mathbb C\)
Index: 5
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Smaller0: \(|w+z|\)
Greater0: \(|w|+|z|\)
Page: 119
Chapter: 4
Section: A
Variables: \(w,z\in\mathbb C\).
Index: 5
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Smaller0: \(|\text{Im }z|\)
Greater0: \(|z|\)
Page: 119
Chapter: 4
Section: A
Variables: \(z\in\mathbb C\).
Index: 5
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Name: A zero of \(p\)
Definition: \(p(\lambda)=0\)
Page: 120, 122
Chapter: 4
Section: A
Variables: \(\mathbb F\) a field
Let \(p\in\mathbb F[x]\)
Index: 7, 11
Definition2: \((z-\lambda)\) divides \(p\)
Conjdef: Some value \(\lambda\in\mathbb F\) such that equivalently
Assuming: nothing

Question: There exists
Answer: unique polynomials \(q,r\in\mathcal P(\mathbb F)\) such that \(p=sq+r\) and \(\text{deg }r<\text{deg }s\).
Page: 121
Chapter: 4
Section: A
Variables: \(\mathbb F\) a field
Let \(p,s\in\mathcal P(\mathbb F)\) with \(s\ne0\).
Index: 8
Name: Division lemma for polynomials

Name: A factor of \(p\)
Definition: A function \(q\in\mathbb F[x]\) such that there exists a polynomial \(s\in\mathcal P(\mathbb F)\) such that \(p=sq\).
Page: 122
Chapter: 4
Section: A
Variables: \(\mathbb F\) a field
\(p\in\mathbb F[x]\)
Index: 10
Assuming: nothing

Question: The number of zeroes of \(P\) is
Answer: at most \(n\).
Page: 123
Chapter: 4
Section: A
Variables: \(P\) be a polynomial of degree \(n\).
Index: 12

Name: An eigenvalue of \(T\)
Definition: there exists \(v\in V\) such that \(v\ne0\) and \(Tv=e v\).
Page: 134, 265
Chapter: 5, 5, 8
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\),
\(V\) a \(\mathbb F\)-vector space
\(T\in\mathcal L(V)\),
Index: 5, 6, 49
Kind: Def,
Definition2: \(T-e I\) is not injective/surjective/invertible (assuming \(V\) finite dimensional)
Conjdef: An element \(e\in\mathbb F\) such that, equivalently
Definition3: \(e\) is a zero of the minimal polynomial of \(T\) (assuming \(V\) is not 0, finite dimensional).
Denoted by: \(\lambda\)
Assuming: nothing

Smaller0: \(p\) is non constant
Greater0: \(p\) has a complex zero
Page: 124
Chapter: 4
Section: A
Variables: \(p\in\mathcal P(\mathbb C)\)
Index: 13
Name: Fundamental theorem of Algebra
French: théorème de d'Alembert(-Gauss)

Name: A factorization of \(p\)
Definition: \(p(z)=c\prod p_i(z)\), with \(c\in\mathbb F\), \(p_i\in\mathcal P(\mathbb F)\) irreduccible.
Page: 125
Chapter: 4
Section: A
Variables: \(\mathbb F\) a field
\(p\in\mathcal P(\mathbb F)\)
Index: 14

Question: The factorizations of \(p\) is
Answer: \[p(z)=c(z-\lambda_1)\dots(z-\lambda_m)\], where \(c,\lambda_1,\dots,\lambda_m\in\mathbb C\), unique.
Page: 125
Chapter: 4
Section: A
Variables: \(p\in\mathcal P(\mathbb C)\)
Index: 14

Smaller0: \((x-\lambda_1)(x-\lambda_2)\) with \(\lambda_1,\lambda_2\in\mathbb R\)
Greater0: \(b^2\ge4c\)
Page: 126
Chapter: 4
Variables: Suppose \(b,c\in\mathbb R\).
Index: 16
Smaller to greater: iff
Hide middle: yes
Prefix Smaller: there is a polynomial factorization of \(x^2+bx+c\) of the form

Question: The set of factorizations of \(p\) is
Answer: a singleton of the form \[p(x)=c(x-\lambda_1)\dots(x-\lambda_m)(x^2+b_1x+c_1)\dots(x^2+b_Mx+c_M)\] with \(b_j^2<4c_j\).
Page: 128
Chapter: 4
Section: A
Variables: \(p\in\mathcal P(\mathbb R)\).
Index: 17

Name: A subspace invariant under \(T\)
Definition: A subspace \(U\subseteq V\) such that \(u\in U\) implies \(Tu\in U\).
Page: 132, 252
Chapter: 5, 8
Section: A, B
Variables: \(\mathbb F\) a field,
\(V\) a \(\mathbb F\)-vector space
\(T\in\mathcal L(V)\)
Index: 2, 3, 21
Kind: Definition, Ex,
Example: \(\{0\}\) and \(V\),
Example2: \(G(\lambda,T)\) for \(\lambda\) an eigenvalue of a complex vector space.
Example3: \(\text{null }T\)
Example4: \(\text{range }T\)

Name: An eigenvector of \(T\) corresponding to \(\lambda\)
Definition: An element \(v\in V\setminus\{0\}\) such that \(Tv=\lambda v\).
Page: 134
Chapter: 5
Section: A
Variables: \(\mathbb F\) a field,
\(V\) a \(\mathbb F\)-vector space
\(T\in\mathcal L(V)\),
\(\lambda\in\mathbb F\)
Index: 7
Kind: Definition
Assuming: nothing

Question: The number of eigenvalues of \(T\) is
Answer: at most dim \(V\).
Page: 136
Chapter: 5
Section: A
Variables: \(\mathbb F\) a field,
\(V\) a \(\mathbb F\)-vector space
\(T\in\mathcal L(V)\)
Index: 13
Assuming: \(V\) finite-dimensional

Smaller0: \(v_1,\dots,v_m\) are corresponding generalized eigenvectors
Greater0: linearly independent
Page: 136, 247
Chapter: 5, 8
Section: A, A
Variables: \(\mathbb F\) a field,
\(V\) a \(\mathbb F\)-vector space,
\(\lambda_1,\dots,\lambda_m\) are distinct eigenvalues of \(T\).
Index: 10, 13
Prefix Greater: \(v_1,\dots,v_m\) are

Name: The restriction operator of \(T\) on the set \(U\)
Definition: An operator in \(\mathcal L(U)\), sending \(u\) to \(Tu\).
Page: 137
Chapter: 5
Section: A
Variables: \(\mathbb F\) a field,
\(V\) a \(\mathbb F\)-vector space
\(T\in\mathcal L(V)\),
\(U\subseteq V\) a subspace invariant under \(T\)
Notation: \(T_{|U}\)
Index: 14
Kind: Definition
Assuming: nothing

Name: The quotient operator
Definition: \((v+U)\mapsto Tv+U\) for \(v\in V\).
Page: 137
Chapter: 5
Section: A
Variables: \(\mathbb F\) a field,
\(V\) a \(\mathbb F\)-vector space
\(T\in\mathcal L(V)\),
\(U\subseteq V\) a subspace
Notation: \(T/U\)
Index: 14
Kind: Definition
Assuming: \(U\) invariant under \(T\)
DefType: \(\mathcal L(V/U)\)

Definition: \(I\) for \(m=0\)
Page: 143
Chapter: 5
Section: B
Variables: \(\mathbb F\) a field,
\(V\) a \(\mathbb F\)-vector space
\(T\in\mathcal L(V)\),
Let \(m\in\mathbb Z\).
Notation: \(T^m\)
Index: 16
Kind: Definition
Definition2: \(T^{m}=T\circ\dots\circ T\) \(m\) times for \(m>0\).
Conjdef: Either
Definition3: \((T^{-1})^{-m}\) asumming \(T\) is invertible, \(m<0\)

Name: \(p(T)\)
Definition: The operator \[\sum_{i=0}^ma_iT^m\].
Page: 143
Chapter: 5
Section: B
Variables: \(\mathbb F\) a field
\(V\) a \(\mathbb F\)-vector space
\(T\in\mathcal L(V)\),
Let \(p\in\mathbb F[x]\).
Index: 17
Kind: Definition
Assuming: nothing

Smaller0: \(V\) is odd-dimensional
Greater0: an eigenvalue
Page: 145, 220, 282
Chapter: 5, 7, 9
Section: A, B, A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space
\(T\in\mathcal{L}(V)\)
Index: 21, 27, 19
Prefix Greater: \(T\) has
Assuming: \(V\) is nonzero finite dimensional
Smaller1: \(\mathbb F=\mathbb C\)
Extra: 7.27 9.19 et si det=0
Connect Smaller0: or
Smaller2: \(T\) is self-adjoint

Name: The diagonal of \(A\)
Definition: \(A_{1,1},\dots,A_{m,m}\).
Page: 147
Chapter: 5
Section: B
Variables: \(\mathbb F\) a field
\(m\in\mathbb N\) and
\(A\in\mathbb F^{m,m}\).
Index: 24
Kind: Definition
Assuming: nothing

Name: An upper triangular matrix
Definition: All the entries below the diagonal of \(M\) are 0
Page: 147, 148
Chapter: 5
Section: B
Variables: \(\mathbb F\) a fi\(V\) be a vector space with basis \(v_1,\dots,v_n\).
Index: 25, 26
Kind: def,
Definition2: \(Tv_j\in \text{span}(v_1,\dots,v_j)\) for each \(j=1,\dots,n\)
Conjdef: A matrix \(M\) defining the linear map \(T\)
Definition3: \(\text{span}(v_1,\dots,v_j)\) is invariant under \(T\) for each \(j=1,\dots,n\).
Definition4: \(M_{i,j}=0\) when \(i>j\).

Definition: \(\mathcal M(T)\) is upper-triangular with respect to some orthonormal basis of \(V\) (Schur's theorem)
Page: 149, 186, 254, 273
Chapter: 5, 8, 8
Section: B, B, D
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
Let \(V\) be a \(\mathbb F\) inner product space,
Let \(T\in\mathcal{L}(V)\).
Index: 27, 38, 23 , 60
Definition2: \((v_1,\dots,v_n)\) is a Jordan basis of \(T\).
Conjdef: There is a basis of \(V\) such that, either
Definition3: \((v_1,\dots,v_n)\) is composed of generalized eigenvectors of \(T\),
Assuming: Let \(V\) non-zero, finite dimensinoal
\(\mathbb F\) is \(\mathbb C\)

Smaller0: \(M\) is upper-triangular
Greater0: \(M\) is inversible
Page: 150
Chapter: 5
Section: B
Variables: \(\mathbb F\) be a field
\(M\) a square matrix
Index: 30
Assuming: Nothing
Smaller1: the elements of the diagonal are nonzero

Smaller0: \(\mathcal M(T,(v_1,\dots,v_n))\) is an upper-triangular matrix
Greater0: the eigenvalues of \(T\)
Page: 152
Chapter: 5
Section: B
Variables: \(\mathbb F\) be a field
\(V\) be a vector space with basis \(v_1,\dots,v_n\)
\(T\in\mathcal L(V)\)
Index: 32
Assuming: Nothing
Greater1: precisely the entries on the diagonal of \(\mathcal M(T,(v_1,\dots,v_n))\)
Connect Greater0: are

Name: A diagonal matrix
Definition: A square matrix such that \(A_{i,j}\ne0\) implies that \(i= j\).
Page: 155
Chapter: 5
Section: C
Variables: \(\mathbb F\) a field
Index: 34
Kind: Definition
Assuming: nothing

Name: The eigenspace of \(T\) corresponding to \(\lambda\)
Definition: \(\text{null }(T-\lambda I)\).
Page: 155
Chapter: 5
Section: C
Variables: \(\mathbb F\) a field
\(V\) be a \(\mathbb F\)-vector space.
\(T\in\mathcal L(V)\)
\(\lambda\in\mathbb F\).
Notation: \(E(\lambda, T)\)
Index: 36
Kind: Definition
Definition2: the set of eigenvectors associated with \(\lambda\)
Assuming: \(\lambda\ne0\)

Name: A diagonalizable map in \(V\)
Definition: the matrix of \(T\) satisfies the same name
Page: 157
Chapter: 5
Section: C
Variables: \(\mathbb F\) a field,
Let \(V\) be a \(\mathbb F\)-vector space.
Index: 41
Definition2: \(V\) has a basis consisting of eigenvalues of \(T\)
Example: Not diagonalizable: \((x,y)\mapsto (y,0)\)
Conjdef: A map \(T\in L(V)\) such that
Definition3: there exists 1-dimensional subspaces \(U_1,\dots,U_n\) of \(V\), each invariant under \(T\), such that \[V=U_1\oplus \dots\oplus U_n.\]
Definition4: \(V=E(\lambda_1,T)\oplus\dots\oplus E(\lambda_m,T)\) for \(\lambda_i\) the distinct eigenvalues
Assuming: \(V\) finite dimensional
Definition5: dim \(V= \text{dim }E(\lambda_1,T)+\dots+\text{dim }E(\lambda_m,T)\) for \(\lambda_i\) the distinct eigenvalues.

Question: The sum of eigenspaces of \(T\) is
Answer: a direct sum.
Page: 156
Chapter: 5
Section: C
Variables: \(\mathbb F\) a field
\(V\) be a \(\mathbb F\)-vector space.
\(T\in\mathcal L(V)\)
Index: 38

Smaller0: \(T\) has \((\text{dim }V)\) distinct eigenvalues
Greater0: \(T\) is diagonalizable
Page: 159
Chapter: 5
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\)-vector space.
\(T\in\mathcal L(V)\).
Index: 44
Extra: \(V\) finite dimensional

Name: The dot product of \((x_1,\dots,x_n), (y_1,\dots,y_n)\)
Definition: \(x_1y_1+\dots+x_ny_n\).
Page: 164
Chapter: 6
Section: A
Variables: \((x_1,\dots,x_n), (y_1,\dots,y_n)\in\mathbb R^n\).
Notation: \(\mathbf x\cdot \mathbf y\)
Index: 2
Kind: Definition

Question: Four properties of dot product
Answer: \(x\cdot x\ge 0\)
Page: 164
Chapter: 6
Section: A
Variables: \(x\in\mathbb R^n\).
Answer2: \(x\cdot x=0\) iff \(x=0\)
Index: 2
Kind: Definition
Answer3: \(\cdot\) is associative
Answer4: the map from \(\mathbb R^n\) to \(\mathbb R\) that sends \(x\in\mathbb R^n\) to \(x\cdot y\) is linear.

Name: Positive binary functon
Definition: A function \(f:V\times V\to \mathbb F\) such that \(f(v,v)\ge0\) for all \(v\in V\).
Page: 166
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\)-vector space
Index: 3
Kind: Definition
Assuming: nothing

Name: A function which have the definiteness property
Definition: A function such that \(\langle v,v\rangle=0\) if and only if \(v=0\).
Page: 166
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\)-vector space
Index: 3
Kind: Definition
Assuming: nothing

Name: Function additive in the first slot
Definition: A binary function \(V\times V\to\mathbb F\) such that \(\langle u+v,w \rangle=\langle u,w \rangle+\langle v,w \rangle\).
Page: 166
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\)-vector space
Index: 3
Kind: Definition
Assuming: nothing

Name: A function satisfying the conjugate symmetry property
Definition: A function \(f:V\times V\to \mathbb F\) such that \(f(u,v)=\overline{f( v,u)}\).
Page: 166
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\)-vector space.
\(\langle\cdot,\cdot\rangle:V\times V\to \mathbb F\) an inner product.
Index: 3
Kind: Definition
Assuming: nothing

Name: Inner product on \(V\)
Definition: Positive,
Page: 166
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\)-vector space.
Index: 3
Kind: Definition
Definition2: definite,
Conjdef: The only binary fonction
Definition3: additive in the first slot,
Definition4: homogene in first slot,
Definition5: conjugate symmetry.

Name: A binary function homogene in the first slot
Definition: A function \(f:V\times V\to \mathbb F\) such that \(f(\lambda u,v)=\lambda f(u,v)\).
Page: 166
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\)-vector space
Index: 3
Kind: Definition
Assuming: nothing

Name: An inner product space
Definition: A vector space \(V\) with an inner product on \(V\).
Page: 167
Chapter: 6
Section: A
Index: 5
Kind: Definition

Page: 167
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) a \(\mathbb F\)-vector space,
\(u\in V\)
Notation: \(v\mapsto\langle v,u\rangle\)
Index: 7
Typ: A linear map from \(V\) to \(\mathbb F\)
Assuming: nothing

Question: \(\langle u,0 \rangle=\langle 0,u \rangle=\)
Answer: \(0\),
Page: 167
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) an \(F\)-inner product space and
\(u\in V\)
Index: 7 bc
Assuming: Nothing

Smaller0: \(\langle u,v+w \rangle\)
Greater0: \(\langle u,v \rangle+\langle u,w \rangle\)
Page: 167
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) an \(F\)-inner product space and
\(u,v,w\in V\)
Index: 7d
Smaller to greater: =
Assuming: Nothing

Question: \(\langle u,\lambda v \rangle=\)
Answer: \(\overline \lambda\langle u,v \rangle\)
Page: 167
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) an \(F\)-inner product space and
\(u,v\in V\)
Index: 7e
Assuming: Nothing

Name: The norm of \(v\)
Definition: \(\sqrt{\langle v,v \rangle }\)
Page: 168
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(v\in V\)
Notation: \(\parallel v\parallel\)
Index: 8
Kind: Definition
Assuming: nothing

Name: Orthogonal vectors
Definition: Two vectors \(u,v\) such that \(\langle u,v \rangle=0\).
Page: 169
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
Let \(u,v\in V\).
Index: 11
Kind: Definition

Question: [$]\norm{\lambda v}[/$] equals
Answer: [$]\abs{\lambda}\norm{v}[/$].
Page: 169
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\),
\(V\) an \(F\)-inner product space and
\(v\in V\), \(\lambda\in\mathbb F\)
Index: 10

Smaller0: \(\langle u,v\rangle\)
Greater0: [$]\norm{u}\norm{v}\cos(\theta)[/$] where \(\theta\) is the angle of \((u,v)\)
Page: 169
Chapter: 6
Section: A
Variables: \(u,v\in\mathbb R^2\).
Index: 11
Smaller to greater: equals
Hide middle: yes
Assuming: \(u,v\) are non 0

Smaller0: [$]\norm{v}=0[/$]
Greater0: \(v=0\)
Page: 169
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(v\in V\)
Index: 10
Smaller to greater: iff

Smaller0: \(\langle v,v \rangle=0\)
Greater0: \(v=0\)
Page: 170
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(v\in V\)
Index: 12
Smaller to greater: iff

Smaller0: \(v=0\)
Greater0: \(v\) is orthogonal to each vector
Page: 170
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(v\in V\)
Index: 12
Smaller to greater: iff

Smaller0: \(u\) is orthogonal to \(v\)
Greater0: [$]\norm{u+v}[/$]
Page: 170
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(v\in V\)
Index: 13
Hide middle: yes
Assuming: Nothing
Greater1: [$]\sqrt{\norm{u}^2+\norm{v}^2} [/$]
Connect Greater0: \(=\)

Page: 171
Implementation: [$]c=\frac{\langle u,v\rangle}{\norm{v}^2}[/$],
\(w=u-cv\)
Section: A
Chapter: 6
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(u,v\in V\)
Name: Orthogonal decomposition proposition
Returns: A vector \(w\) and a scalar \(c\) such that:
-\(\langle w,v\rangle=0\) and
-\(u=cv+w\)
Index: 14
Assuming: \(v\neq 0\)

Smaller0: [$]\abs{\langle u,v\rangle}[/$]
Greater0: [$]\norm{u}\norm{v}[/$]
Page: 172
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) a \(\mathbb F\)-inner product space.
\(u,v\in V\)
Index: 15
Smaller to greater: \(\le\)
Assuming: Nothing
Name: The Cauchy-Schwarz inequality
Greater to smaller: \(\ge\)

Smaller0: One of the vector is a scalar multiple of the other
Greater0: [$$]\abs{\langle u,v \rangle}=\norm{u}\norm{v}[/$$]
Page: 172
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\),
\(V\) an \(F\)-inner product space and
\(u,v\in V\)
Index: 15
Hide middle: yes
Assuming: Nothing

Name: Triangle inequality
Definition: [$]\norm{u+v}\le\norm{u}+\norm{v}[/$]
Page: 173
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(u,v\in V\)
Index: 18
Assuming: nothing

Smaller0: [$]\norm{u+v}^2+\norm{u-v}^2[/$]
Greater0: [$]2(\norm{u}^2+\norm{v}^2)[/$]
Page: 174
Chapter: 6
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(u,v\in V\)
Index: 22
Smaller to greater: =
Assuming: Nothing
Name: Parallelogram Equality

Name: An orthonormal list of vectors
Definition: A list of vectors such that each vector in the list has norm 1 and is orthogonal to all the other vectors in the list.
Page: 180
Chapter: 6
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space
Index: 23
Kind: Definition
Assuming: nothing

Smaller0: \(v_1,\dots,v_m\) is an orthonormal list of vectors in \(V\)
Greater0: [$]\norm{\sum_{i=1}^ma_iv_i}^2[/$]
Page: 180
Chapter: 6
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(v_1,\dots,v_m\in V\)
Index: 25
Hide middle: yes
Greater1: \(\sum_{i=1}^m|a_i|^2 \)
Connect Greater0: =
Suffix Greater: for all \(a_1,\dots,a_m\in\mathbb F\)

Smaller0: \((a_1,\dots,a_m)\) is orthonormal
Greater0: \((a_1,\dots,a_m)\) is a basis of \(V\)
Page: 181
Chapter: 6
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
Let \(V\) be a \(\mathbb F\) inner product space.
Let \((a_1,\dots,a_m)\)
Index: 28
Assuming: \(V\) finite-dimensional
Smaller1: \(m=\text{dim}(V)\)
Connect Smaller0: and

Name: Orthonormal basis of \(V\)
Definition: A basis of \(V\) which is an orthonormal list.
Page: 181
Chapter: 6
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space
Index: 27
Kind: Definition
Assuming: nothing

Smaller0: \(v_1,\dots,v_m\) is an orthonormal basis of vectors in \(V\)
Greater0: \(w\)
Page: 182
Chapter: 6
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(v_1,\dots,v_m,w\in V\)
Index: 30
Hide middle: yes
Assuming: Nothing
Greater1: \(\langle w,v_i\rangle v_i\)
Connect Greater0: =\(\sum_{i=1}^n\)
Greater2: [$]\norm{v}^2[/$]
Connect Greater1: and
Greater3: [$]\sum_{i=1}^n\abs{\langle w,v_i \rangle}^2[/$]
Connect Greater2: =

Page: 183
Implementation: [$$]w_j=\frac{v_j-\sum_{i=1}^{j-1}\langle v_j,w_i \rangle w_i}{\norm{v_j-\sum_{i=1}^{j-1}\langle v_j,w_i \rangle w_i}}.[/$$]
Section: B
Chapter: 6
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
Let \(V\) be a \(\mathbb F\) inner product space.
\(v_1,\dots,v_m\in V\)
Name: Gram-Schmidt Procedure
Returns: \(w_1,\dots,w_m\) is an orthonormal list of vectors in \(V\) such that \(\text{span}(v_1,\dots,v_j)=\text{span}(w_1,\dots,w_j)\) for \(j=1,\dots,m\).
Index: 31
Assuming: \(v_1,\dots,v_m\) is linearly independant list of vectors in \(V\).

Question: \(V\) admits
Answer: an orthonormal basis.
Page: 185
Chapter: 6
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space
Index: 34
Assuming: \(V\) finite dimensional

Smaller0: \(v_1,\dots,v_n\) is an orthonomal list of vectors
Greater0: \(v_1,\dots,v_n\) can be extended to an orthonormal basis of \(V\)
Page: 185
Chapter: 6
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(v_1,\dots,v_n\in V\)
Index: 35
Smaller to greater: iff
Assuming: Nothing

Name: Riesz Representation theorem
Definition: A vector \(u\in V\) such that \(\phi(v)=\langle v,u \rangle\) for every \(v\in V\).
Page: 188
Chapter: 6
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(\phi\) is a linear functional on \(V\)
Index: 42
Assuming: Nothing
Cardinal: unique

Name: Orthogonal complement of \(U\)
Definition: The set of all vectors in \(V\) that are orthogonal to every vector in \(U\):
\[\{v\in V: \langle v,u \rangle=0\text{ for every }u\in U\}.\]
Page: 193
Chapter: 6
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(U\) a subset of \(V\)
Notation: \(U^\bot\)
Index: 45
Kind: Definition
Typ: A subspace of \(V\)
Assuming: Nothing

Smaller0: \(U\subseteq W\)
Greater0: \(\supseteq\)
Page: 193
Chapter: 6
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(U,W\) subsets of \(V\)
Index: 46
Hide middle: yes
Prefix Greater: \(W^\bot\)
Assuming: Nothing
Extra: reformuler
Suffix Greater: \(U^\bot\)

Definition: The function sending \(v\) to the only \(u\in U\) such that \(v-u\in U^\bot\).
Page: 193
Section: C
Chapter: 6
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space
\(U\) a subspace of \(V\)
Notation: \(P_U\)
Name: The operator of orthogonal projection of \(V\) onto \(U\)
Kind: 53 55
Linear: \(\mathcal L(V)\)
Assuming: \(U\) finite-dimensional

Question: \(V^\bot=\)
Answer: \(\{0\} \)
Page: 193
Chapter: 6
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space
Index: 46
Assuming: Nothing

Smaller0: \(U\cap U^\bot\)
Greater0: \(\{0\}\)
Page: 193
Chapter: 6
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space
\(U\) a subspace of \(V\)
Index: 46
Smaller to greater: \(\subseteq\)
Assuming: Nothing
Greater to smaller: \(\supseteq\)

Question: \(\{0\}^\bot=\)
Answer: \(V\)
Page: 193
Chapter: 6
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space
Index: 46
Assuming: Nothing

Question: \(\text{dim }U^\bot=\)
Answer: \(\text{dim }V-\text{dim }U\).
Page: 195
Chapter: 6
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space
\(U\) a subspace of \(V\)
Index: 50
Assuming: \(V\) finite-dimensional

Question: \((U^\bot)^\bot=\)
Answer: \(U\).
Page: 195
Chapter: 6
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space
\(U\) a subspace of \(V\)
Index: 51
Assuming: \(U\) finite dimensional

Definition: \(v-P_Uv\)
Page: 196
Chapter: 6
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(U\) a subspace of \(V\),
\(v\in V\)
Index: 55 f
Typ: element of \(U^\bot\)
Assuming: \(U\) finite-dimensional

Smaller0: \(v\in U\)
Greater0: \(v\)
Page: 196
Chapter: 6
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(U\) a subspace of \(V\),
\(v\in V\)
Index: 55 a
Smaller to greater: iff
Prefix Greater: \(P_Uv=\)
Assuming: \(U\) is a finite-dimensional

Smaller0: \(v\in U^\bot\)
Greater0: \(0\)
Page: 196
Chapter: 6
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space
\(U\) a subspace of \(V\)
Index: 55 c
Prefix Greater: \(P_Uv=\)
Assuming: \(U\) finite dimensional

Question: \(\text{range}P_U\)
Answer: equals \(U\)
Page: 196
Chapter: 6
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(U\) a subspace of \(V\),
\(v\in V\)
Index: 55 d
Assuming: \(U\) finite-dimensional

Smaller0: [$]\norm{P_Uv}[/$]
Greater0: [$]\norm{v}[/$]
Page: 196
Chapter: 6
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(U\) a subspace of \(V\),
\(v\in V\)
Index: 55 h
Smaller to greater: \(\le\)
Assuming: \(U\) finite-dimensional
Greater to smaller: \(\ge\)

Smaller0: \(w_1,\dots,w_m\) is an orthonormal basis of \(U\)
Greater0: \(\sum_{i=1}^m\langle v,w_i \rangle w_i\)
Page: 196
Chapter: 6
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(U\) a subspace of \(V\),
\(v,w_1,\dots,w_m\in V\)
Index: 55 i
Hide middle: yes
Assuming: \(U\) finite-dimensional,
Greater1: \(P_Uv\)
Connect Greater0: =

Question: \(P_U^2\) equals
Answer: \(P_U\).
Page: 196
Chapter: 6
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
Let \(V\) be a \(\mathbb F\) inner product space.
\(U\) a subspace of \(V\),
Index: 55 g
Assuming: \(U\) finite-dimensional

Smaller0: \(u=P_Uv\)
Greater0: [$]\norm{v-P_Uv}=\norm{v-u}[/$]
Page: 198
Chapter: 6
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
Let \(V\) be a \(\mathbb F\) inner product space.
\(U\) a subspace of \(V\),
\(v\in V\), \(u\in U\)
Index: 56
Smaller to greater: iff
Assuming: \(U\) finite-dimensional

Smaller0: [$]\norm{v-P_Uv}[/$]
Greater0: [$]\norm{v-u}[/$]
Page: 198
Chapter: 6
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
Let \(V\) be a \(\mathbb F\) inner product space.
\(U\) a subspace of \(V\),
\(v\in V\), \(u\in U\)
Index: 56
Smaller to greater: \(\le\)
Assuming: \(U\) finite-dimensional
Greater to smaller: \(\ge\)

Name: The adjoint of \(T\)
Definition: A function \(S\) such that \(\langle Tv,w \rangle=\langle v,Sw \rangle\).
Page: 204
Chapter: 7
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V,W\) be a \(\mathbb F\) inner product space.
\(T\in\mathcal{L}(V,W)\)
\(v\in V\), \(w\in W\)
Notation: \(T^*\)
Index: 2
Kind: Definition
DefType: \(\mathcal L(W,V)\)

Question: \((T^*)^*=\)
Answer: \(T\)
Page: 206
Chapter: 7
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V,W\) be a \(\mathbb F\) inner product space.
\(T\in\mathcal{L}(V,W)\)
Index: 6c
Assuming: Nothing

Smaller0: \((S+T)^*\)
Greater0: \(S^*+T^*\)
Page: 206
Chapter: 7
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V,W\) be a \(\mathbb F\) inner product space.
\(S,T\in\mathcal{L}(V,W)\)
Index: 6a
Smaller to greater: =
Hide middle: yes
Assuming: Nothing

Smaller0: \((\lambda T)^*\)
Greater0: \(\overline\lambda T^*\)
Page: 206
Chapter: 7
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V,W\) be a \(\mathbb F\) inner product space.
\(T\in\mathcal{L}(V,W)\)
\(\lambda\in\mathbb F\)
Index: 6b
Smaller to greater: =
Hide middle: yes
Assuming: Nothing

Question: \(I^*=\)
Answer: \(I\)
Page: 206
Chapter: 7
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(T\in\mathcal{L}(V)\)
Index: 6d
Assuming: nothing

Smaller0: \((ST)^*\)
Greater0: \(T^*S^*\)
Page: 206
Chapter: 7
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V,W\) be a \(\mathbb F\) inner product space.
\(S,T\in\mathcal{L}(V,W)\)
Index: 6e
Smaller to greater: =
Hide middle: yes
Assuming: Nothing

Smaller0: \(\text{range }T^*\)
Greater0: \((\text{null }T)^\bot\)
Page: 207
Chapter: 7
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V,W\) be a \(\mathbb F\) inner product space.
\(T\in\mathcal{L}(V,W)\)
Index: 7 a
Smaller to greater: =
Assuming: nothing

Name: The conjugate transpose of \(A\)
Definition: the conjugate of the transpose of \(A\).
Page: 208
Chapter: 7
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(A\in\mathbb F^{m\times n}\).
Index: 8
Assuming: nothing

Smaller0: \(\mathcal M(T,(v_1,\dots,v_n),(w_1,\dots,w_m))\)
Greater0: \(\mathcal M(T^*,(w_1,\dots,w_m),(v_1,\dots,v_n))\)
Page: 208
Chapter: 7
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V,W\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\) and \(w_1,\dots,w_m\).
\(T\in\mathcal{L}(V,W)\)
Index: 10
Smaller to greater: equals
Hide middle: yes
Assuming: \(v_1,\dots,v_n\) and \(w_1,\dots,w_m\) are orthonormal
Prefix Smaller: The conjugate transpose of

Name: Self-adjoint
Definition: \(T=T^*\).
Page: 209, 211
Chapter: 7
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space
\(v_1,\dots,v_n\) a basis
Index: 11, 15
Kind: Definition
Definition2: for all \(v\in V\), \(\langle Tv,v\rangle\in\mathbb R\) (assuming \(\mathbb F=\mathbb C\))
Conjdef: \(T\in\mathcal{L}(V)\) such that equivalently
Definition3: \(V\) has an orthonormal basis consisting of eigenvectors of \(T\) (assuming \(\mathbb{F=R}\), finite dimensional)
Name2: Hermitian
Extra: Real spectral theorem
Definition4: \(V\) has an orthonormal basis such that \(T\) is diagonal (assuming \(\mathbb{F=R}\), finite dimensional)

Question: Self-adjoint map is similar to
Answer: real numbers.
Page: 209
Chapter: 7
Section: A
Index: 12
Kind: ex

Question: Adjoint map is similar to
Answer: conjucate of complex number.
Page: 209
Chapter: 7
Section: A
Index: 12
Kind: ex

Name: The class of self-adjoint matrices
Page: 209
Section: A
Chapter: 7
Index: 12
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\)
\(T,S\in\mathcal{L}(V)\)
Kind: Example
Under: addition

Smaller0: \(V\) is a complex space
Greater0: \(T=0\)
Page: 210
Chapter: 7
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\).
\(T\in\mathcal{L}(V)\).
Index: 14 16
Hide middle: yes
Assuming: nothing
Smaller1: \(T\) is self-adjoint
Prefix Smaller: (
Connect Smaller0: or
Smaller2: (for all \(v\), \(\langle Tv,v \rangle=0\))
Connect Smaller1: ) and

Smaller0: \(T\) is self-adjoint
Greater0: reals
Page: 210
Chapter: 7
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\).
\(T\in\mathcal{L}(V)\).
Index: 13
Prefix Greater: the eigenvalues of \(T\) are
Assuming: Nothing

Name: A normal map
Definition: \(TT^*=T^*T\).
Page: 212, 218, 291
Chapter: 7, 9
Section: A, B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
Index: 18 20, 24
Kind: Definition
Definition2: [$]\norm{Tv}=\norm{T^*v}[/$]
Conjdef: \(T\in\mathcal{L}(V)\) such that equivalently
Definition3: \(V\) has an orthonormal basis consisting of eigenvectors of \(T\) (assuming \(\mathbb{F=C}\), finite dimensional)
Extra: Complex Spectral Theorem
Definition4: \(T\) has a diagonal matrix with respect to some orthonormal basis of \(V\) (assuming \(\mathbb{F=C}\), finite dimensional)
Definition5: there is an orthonormal basis of \(V\) with respect to which \(T\) has a block diagonal matrix such that each block is a 1-by-1 matrix or a 2-by-2 matrix of the form \(\left(\begin{array}{ll}a&-b\\b&a\end{array}\right)\) with \(b>0\) (for \(\mathbb{F=R}\) finite dimensional)

Smaller0: \(T\) is normal
Greater0: \(\text{null }T\)
Page: 212
Chapter: 7
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(T\in\mathcal{L}(V)\).
Index: 19
Kind: Example
Hide middle: yes
Prefix Greater: \(\text{null }T^*=\)
Assuming: nothing

Smaller0: \(T\) is normal
Greater0: the eigenvector of \(T\) with eigenvalue \(\overline\lambda\)
Page: 213
Chapter: 7
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space.
\(T\in\mathcal{L}(V)\).
Index: 21
Prefix Greater: the eigenvectors of \(T^*\) with eigenvalue \(\lambda\) are
Assuming: nothing

Question: The eigenvectors of \(T\) corresponding to distinct eigenvalues are
Answer: orthogonal.
Page: 214
Chapter: 7
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\)
\(T\in\mathcal{L}(V)\)
Index: 22
Assuming: Nothing

Question: Difference between real and complex spectral theorem:
Answer: self adjoint for real, and normal for complex.
Page: 218, 221
Chapter: 7
Section: B
Index: 24, 29

Smaller0: \(T\) is self-adjoint
Greater0: invertible
Page: 219
Chapter: 7
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\)
\(T\in\mathcal{L}(V)\)
\(b,c\in\mathbb R\)
Index: 26
Prefix Greater: \(T^2+bT+cI\) is
Assuming: nothing
Smaller1: \(b^2<4c\)

Name: "\(U\) is subspace invariant by \(T\)"
Page: 221, 288
Section: B, B
Chapter: 7, 9
Index: 28, 30
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space
\(U\subseteq V\)
Under: orthogonal complement
Condition2: \(T\) self-adjoint
Under3: taking the adjoint of \(T\)
Condition3: normal on real vector space
Under4: product
Condition4: ?

Question: Self-adjoint are:
-closed under restriction of \(T\) to
Answer: \(T\)-invariant subspace of \(V\)
Page: 221
Chapter: 7
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
Let \(V\) be a \(\mathbb F\) inner product space
Index: 28

Name: A positive operator
Page: 225
Chapter: 7
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space
\(T\in\mathcal{L}(V)\)
Index: 31

Name: A positive operator
Definition: \(\langle Tv,v\rangle\ge0\) for all \(v\in V\).
Page: 225, 226
Chapter: 7
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\)
\(T\in\mathcal{L}(V)\)
Index: 31, 35
Definition2: \(T\) is self-adjoint and all the eingevalues of \(T\) are nonnegative
Conjdef: An operator \(T\) such that, equivalently
Definition3: \(T\) has a positive square root,
Definition4: \(T\) has a self-adjoint square root,
Assuming: \(V\) finite dimensional
Definition5: There exists an operator \(R\in\mathcal L(V)\) such that \(T=R^*R\).

Smaller0: \(T\) is positive
Greater0: exactly one
Page: 227
Chapter: 7
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\)
\(T\in\mathcal{L}(V)\)
Index: 36
Prefix Greater: the number of positive square root of \(T\) is
Assuming: \(V\) finite dimensional

Name: An unitary operator
Definition: An operator which is an isometry.
Page: 228
Chapter: 7
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
Index: 41
Kind: Example
Assuming: \(\mathbb F=\mathbb C\)

Name: An orthogonal operator
Definition: An isometry
Page: 229
Chapter: 7
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
Context: Analysis
Index: 42
Assuming: \(\mathbb F=\mathbb R\)

Name: An isometry
Definition: \(||Tv||=||v||\) for all \(v\).
Page: 228, 229, 231
Chapter: 7
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space
Index: 37, 42, 43
Kind: def
Definition2: \(\langle Tu,Tv \rangle=\langle u,v\rangle\) for all \(u,v\).
Conjdef: An element \(T\in\mathcal L(v)\) such that, equivalently
Definition3: there exists/for all orthonormal basis \(e_1,\dots,e_n\), \(Te_1,\dots,Te_n\) is also an orthonormal basis (assuming finite-dimensional)
Definition4: \(T^*T=TT^*=I\), that is \(T^{-1}\) exists and equals \(T^*\), (assuming there is an orthonormal basis)
Assuming: \(V\) finite dimensional
Definition5: there is an orthonormal basis of \(V\) consisting of eigenvectors of \(S\) whose corresponding eigenvalues all have absoluve value 1 (assuming \(\mathbb{F=C}\) there is an orthonormal basis)
Definition6: there exists an orthonormal basis of \(V\) with respect to which \(T\) has a block diagonal matrix such that each block on the diagonal is a 1-by-1 matrix containing 1 or -1 or is a 2-by-2 matrix of the form \(\left(\begin{array}{ll}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{array}\right)\) with \(\theta\in(0,\pi)\) (\(\mathbb{F=R}\) finite dimensional)

Name: Isometries
Page: 229 https://en.wikipedia.org/wiki/Isometry
Section: C
Chapter: 7
Under: composition
Under2: adjoint
Under3: inverse (always exists)

Definition: The unique positive square root of \(T\).
Page: 233
Chapter: 7
Section: D
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space
\(T\in\mathcal{L}(V)\)
Notation: \(\sqrt T\)
Index: 44
Kind: Notation
Assuming: \(V\) finite dimensional
\(T\) is a positive operator

Name: Polar decomposition of \(T\)
Definition: \(T\) written as \(S\sqrt{T^*T}\), with \(S\in\mathcal L(V)\) an isometry
Page: 233
Chapter: 7
Section: D
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\)
\(T\in\mathcal{L}(V)\)
Index: 45
Assuming: \(V\) finite dimensional
Construction: \(S\) can be defined by taking orthonormal bases \(e_1,\dots,e_m\) of \(\left(\text{range} T\right)^\bot\), \(f_1,\dots,f_m\) of \(\left(\text{range} (\sqrt{T^*T})\right)^\bot\), and for \(u=v+\sum a_ie_i\) with \(v\in\text{range} (\sqrt{T^*T})\), \(Uv= T\left(\sqrt{T^*T}^{-1}\right)v+\sum a_if_i\).

Name: The singular values of \(T\)
Definition: The eigenvalues of \(\sqrt{T^*T}\), with each eigenvalue \(\lambda\) repeated \(\text{dim }E(\lambda,\sqrt{T^*T})\) times
Page: 236
Chapter: 7
Section: D
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
Let \(V\) be a \(\mathbb F\) inner product space.
Let \(T\in\mathcal{L}(V)\).
Index: 49, 52
Kind: Definition
Definition2: The nonnegative square roots of the eigenvalues of \(T^*T\), with each eigenvalue \(\lambda\) repeated \(\text{dim }E(\lambda,T^*T)\) times.

Page: 237
Implementation: The value of \(u_i\) can be taken to be \(S t_i\) where \(T=S\sqrt{T^*T}\).
Section: D
Chapter: 7
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\)
\(T\in\mathcal{L}(V)\)
Name: Singular value decomposition
Returns: \(t_1,\dots,t_n\) and \(u_1,\dots,u_n\) of \(V\) such that \[Tv=\sum_{i=1}^{n} s_i\langle v,t_i\rangle u_i\] for every \(v\in V\).
Index: 51
Typ: orthonormal bases
Assuming: Suppose \(T\in\mathcal L(V)\) has singular values \(s_1,\dots,s_n\)

Smaller0: \(\text{null }T^{i}=\text{null }T^{i+1}\)
Greater0: for all \(n>i\), \(\text{null }T^{i}=\text{null }T^{n}\)
Page: 242
Chapter: 8
Section: A
Variables: \(T\in\mathcal{L}(V)\).
Index: 3
Smaller to greater: iff
Assuming: nothing

Question: \[\text{null }T^0=\]
Answer: \[\{0\} \].
Page: 242
Chapter: 8
Section: A
Variables: \(\mathbb F\) a field,
\(T\in\mathcal{L}(V)\).
Index: 2
Assuming: nothing

Smaller0: \(\text{null }T^i\)
Greater0: \(\text{null }T^{i+1}\)
Page: 242
Chapter: 8
Section: A
Variables: \(\mathbb F\) a field
\(V\) an \(F\)-inner product space and
\(T\in\mathcal{L}(V)\).
Index: 2
Smaller to greater: \(\subseteq\)
Assuming: \(V\) finite-dimensional
Greater to smaller: \(\supseteq\)

Question: \(\text{null }T^{\text{dim }V+1}=\)
Answer: \(\text{null }T^{\text{dim }V} \).
Page: 243
Chapter: 8
Section: A
Variables: \(\mathbb F\) a field
Let \(T\in\mathcal{L}(V)\).
Index: 4
Assuming: \(V\) finite dimensional

Smaller0: \(V\)
Greater0: \(\text{null }T^{\text{dim }V}\)
Page: 243, https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem
Chapter: 8
Variables: \(\mathbb F\) a field
\(T\in\mathcal{L}(V)\)
Index: 5
Kind: A
Smaller to greater: \(=\)
Assuming: \(V\) finite dimensional
Name: Rank-nullity theorem
Greater1: \(\text{range }T^{\text{dim }V} \)
Connect Greater0: \(\oplus\)
French: Théorème du rang

Name: A generalized eigenvector of \(T\) of rank \(m\) corresponding to \(\lambda\)
Definition: A vector \(v\ne0\) such that \((T-\lambda I)^mv=0\) and \((T-\lambda I)^{m-1}v=0\)
Page: 245, https://en.wikipedia.org/wiki/Generalized_eigenvector
Chapter: 8
Section: A
Variables: \(\mathbb F\) a field
Let \(V\) be a \(\mathbb F\) vector space
Let \(T\in\mathcal{L}(V)\).
\(\lambda\) an eigenvalue of \(T\)
\(m\in\mathbb N\)
Index: 9
Kind: Definition

Name: A generalized eigenvector of \(T\) corresponding to \(\lambda\)
Definition: A generalized eigenvector of \(T\) of rank \(m\) corresponding to \(\lambda\) for some \(m\ge0\).
Page: 245
Chapter: 8
Section: A
Variables: \(\mathbb F\) a field
Let \(V\) be a \(\mathbb F\) vector space
Let \(T\in\mathcal{L}(V)\).
\(\lambda\) an eigenvalue of \(T\)
Index: 9
Kind: Definition
Assuming: finite dimensional

Name: The generalized eigenspace of \(T\) corresponding to \(\lambda\)
Definition: The set of all generalized eigenvectors of \(T\) corresponding to \(\lambda\), along with the 0 vector.
Page: 245, 246, 252
Chapter: 8
Section: A, B
Variables: \(\mathbb F\) a field
\(V\) be a \(\mathbb F\) vector space
\(T\in\mathcal{L}(V)\).
\(\lambda\) an eigenvalue of \(T\)
Notation: \(G(\lambda,T)\)
Index: 10
Kind: Definition
Definition2: \(\text{null }(T-\lambda I)^{\text{dim }V}\)
Assuming: \(V\) is nonzero, finite-dimensional
Property: Invariant under \(T\)

Name: A nilpotent operator
Definition: An operator such that some power of it equals 0.
Page: 248
Chapter: 8
Section: A
Variables: \(\mathbb F\) a field
Let \(V\) be a \(\mathbb F\) vector space
Let \(T\in\mathcal{L}(V)\).
Index: 16

Smaller0: \(T\) is nilpotent
Greater0: the matrix of \(T\) is upper-triangle with a 0 diagonal
Page: 249
Chapter: 8
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\)
\(T\in\mathcal{L}(V)\)
Index: 18 19
Hide middle: yes
Prefix Greater: the basis can be choosen such that
Assuming: \(V\) is nonzero finite-dimensional
Greater1: \(T^{\text{dim }V}=0\)
Connect Greater0: and

Question: The \(G(\lambda_i,T)\)'s
Answer: are in direct sum and generate \(V\)
Page: 252
Chapter: 8
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\),
\(V\) an \(F\)-vector space and
\(T\in\mathcal{L}(V)\).
\(\lambda_1,\dots,\lambda_m\)'s be the distinct eigenvalues of \(T\).
Index: 21 a
Assuming: \(V\) finite-dimensional nonzero,
\(\mathbb F=\mathbb C\)

Question: \((T-\lambda I)_{|G(\lambda,T)}\), for \(\lambda\) an eigenvalue of \(T\)
Answer: is nilpotent
Page: 252
Chapter: 8
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) vector space
\(T\in\mathcal{L}(V)\).
Index: 21c
Assuming: \(V\) is nonzero finite-dimensional,
\(\mathbb{F=C}\)

Smaller0: sum of the multiplicities
Greater0: \(\text{dim } V\)
Page: 255
Chapter: 8
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\)
\(T\in\mathcal{L}(V)\)
Index: 26
Smaller to greater: equals
Hide middle: yes
Assuming: \(V\) is nonzero finite-dimensional
Prefix Smaller: the
Suffix Smaller: of all eigenvalues of \(T\)

Name: The multiplicity of \(\lambda\)
Definition: The dimension of the corresponding generalized eigenspace \(G(\lambda,T)\)
Page: 254
Chapter: 8
Variables: \(\mathbb F\) be a field
\(V\) be a vector space
\(T\in\mathcal{L}(V)\)
\(\lambda\) an eigenvalue of \(T\)
Index: 24
Kind: Definition
Definition2: \(\left(\text{dim null }(T-\lambda I)^{\text{dim }V}\right)\).
Name2: The algebraic multiplicity of \(\lambda\)
Extra: degree of lambda as a zero
Assuming: \(V\) is nonzero finite-dimensional

Name: Geometric multiplicity of \(\lambda\)
Definition: \(\left(\text{dim null }(T-\lambda I)\right)\).
Page: 254
Chapter: 8
Variables: \(\mathbb F\) be a field
\(V\) be a vector space
\(T\in\mathcal{L}(V)\)
\(\lambda\) an eigenvalue of \(T\)
Index: 24
Kind: Definition

Question: The degree of the minimal polynomial of \(T\) is
Answer: at most \(\text{dim }V\).
Page: 255
Chapter: 9
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\),
\(V\) an \(F\)-inner product space and
\(T\in\mathcal{L}(V)\).
Index: 26
Assuming: Finite dimensional

Name: A block diagonal matrix
Definition: A square matrix of the form
\[\left(\begin{array}{lll} A_1&&0\\ &\dots&\\0&&A_m\end{array}\right)\]
where \(A_1,\dots,A_m\) are square matrices lying along the diagonal and the other entries of the matrix equal 0.
Page: 255
Chapter: 8
Variables: The (algebraic) multiplicity of \(\lambda\)
Index: 27
Kind: Definition

Question: There is a basis of \(V\) with respect to which \(T\) has a matrix of the form:
Answer: block diagonal matrix
\[\left(\begin{array}{lll}A_1&&0\\&\dots&\\0&&A_m\end{array}\right)\]
where each \(A_j\) is a \(d_j\)-by-\(d_j\) upper-triangular matrix of the form
\[\left(\begin{array}{lll}\lambda_j&&*\\&\dots&\\0&&\lambda_j\end{array}\right)\] with \(*\) arbitrary entries
.
Page: 256
Chapter: 8
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
Let \(V\) be a \(\mathbb F\) inner product space.
Let \(T\in\mathcal{L}(V)\).
Let \(\lambda_1,\dots,\lambda_m\) be the distinct eigenvalues of \(T\), with multiplicieties \(d_1,\dots,d_m\).
Index: 29
Assuming: \(V\) is nonzero finite-dimensional,
\(\mathbb F\) is \(\mathbb C\)

Smaller0: \(T\) is nilpotent
Greater0: \(I+T\)
Page: 258
Chapter: 8
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product.
\(T\in\mathcal{L}(V)\).
Index: 31
Kind: B
Hide middle: yes
Assuming: \(V\) is nonzero finite-dimensional
Greater1: a square root
Connect Greater0: has
Construction: ?

Smaller0: \(T\) is invertible
Greater0: a square root
Page: 259, 308
Chapter: 8, 10
Section: B
Variables: \(\mathbb F\) a field
\(V\) be a \(\mathbb F\) inner product space
\(T\in\mathcal{L}(V)\).
Index: 33, 24
Prefix Greater: (\(\mathbb F\) is \(\mathbb C\) implies \(T\) has
Assuming: \(V\) is nonzero finite-dimensional
Greater1: is nonzero
Connect Greater0: ) and \(\text{det }T\)
Extra: plusieurs then?
Counterexample: square 0 matrix

Smaller0: \(\lambda\) is a zero of the characteristic polynomial of \(T\)
Greater0: \(\lambda\) is an eigenvalues of \(T\)
Page: 261
Chapter: 8
Section: C
Variables: \(\mathbb F\) a field
\(V\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\).
\(T\in\mathcal{L}(V)\).
\(\lambda\in V\)
Index: 36 b
Smaller to greater: iff
Hide middle: yes
Assuming: \(V\) is nonzero
Extra: 9.24

Name: the characteristic polynomial of \(T\)
Definition: The polynomial \(\Pi(z-\lambda_i)^{d_i}\) with \(\lambda_1,\dots,\lambda_m\) be the distinct eigenvalues of \(T\), with multiplicities \(d_1,\dots,d_m\).
Page: 261
Chapter: 8
Section: B
Variables: \(\mathbb F\) a field
Let \(V\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\).
Let \(T\in\mathcal{L}(V)\)
Index: 34
Kind: Definition
Definition2: \(\text{det}(zI-T)\).
Extra: 10.25
Assuming: \(V\) is nonzero finite-dimensional

Question: \(q(T)\) equals
Answer: 0
Page: 261
Chapter: 8, 9
Section: C
Variables: \(\mathbb F\) a field
\(V\) be a \(\mathbb F\) inner product space.
\(T\in\mathcal{L}(V)\)
\(q\) its characteristic polynomial.
Index: 37, 24
Assuming: \(V\) is nonzero finite-dimensional
Name: Cayley-Hamilton Theorem

Question: The characteristic polynomial of \(T\) has degree
Answer: \(\text{dim }V\).
Page: 261
Chapter: 8
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\),
\(V\) an \(F\)-inner product space and
\(T\in\mathcal{L}(V)\).
Index: 36 a
Assuming: Nothing
Extra: 9.24

Name: A monic polynomial
Definition: A polynomial whose highest-degree coefficient equals 1.
Page: 262
Chapter: 8
Section: C
Index: 38

Name: The minimal polynomial of \(T\)
Definition: The unique monic polynomial \(p\) of smallest degree such that \(p(T)=0\).
Page: 262, 263
Chapter: 8., 9
Section: C
Variables: \(\mathbb F\) a field
\(V\) a vector space
\(T\in\mathcal{L}(V)\)
Index: 40, 43, 26
Assuming: Let \(V\) non-zero finite dimensional

Smaller0: the characteristic polynomial of \(T\)
Greater0: the minimal polynomial of \(T\)
Page: 265
Chapter: 8
Section: C
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(V\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\)
\(T\in\mathcal{L}(V)\)
Index: 48
Smaller to greater: is a multiple of
Assuming: Let \(V\) non-zero finite dimensional
Greater to smaller: divides
Extra: 9.26

Name: Jordan basis for \(T\)
Definition: A basis of \(V\) such that, with respect to this basis, \(T\) has a block diagonal matrix \[\left(\begin{array}{lll}A_1&&0\\&\dots&\\0&&A_m\end{array}\right)\]
where each \(A_j\) is a \(d_j\)-by-\(d_j\) upper-triangular matrix of the form
\[\left(\begin{array}{llll}\lambda_j &1 & &0 \\ &\dots&\dots & \\ & &\dots &1\\ 0 & & &\lambda_j\end{array}\right)\]
.
Page: 273
Chapter: 8
Section: D
Variables: \(\mathbb F\) a field,
Let \(V\) be a \(\mathbb F\) vector space
Let \(T\in\mathcal{L}(V)\).
Index: 59
Assuming: Let \(V\) non-zero finite dimensional

Page: 276
Chapter: 9
Section: A
Name: The complexification of \(V\)
Set: its elements are the ordered pairs \((u,v)\), where \(u,v\in V\), (written \(u+iv\)).
Variables: \(V\) be a \(\mathbb R\) vector space
\(T\in\mathcal{L}(V)\).
Notation: \(V_{\mathbb C}\)
Kind: Definition
Index: 2
+: Addition is defined by \((u+iv)+(u'+iv')=(u+u')+i(v+v')\).
0: 0+i0
×: Complex scalar multiplication is defined by \[(a+bi)(u+iv)=(au-bv)+i(av+bu)\] for \(a,b\in\mathbb R\) and \(u,v\in V\).
Base: any base of V
Commutative: y
Field-base: \(\mathbb C\)

Name: The complexification of \(T\)
Definition: \[(u+iv)\mapsto Tu+iTv\] for \(u,v\in V\).
Page: 277
Chapter: 9
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) vector space
\(T\in\mathcal{L}(V)\).
Notation: \(T_{\mathbb C}\)
Index: 5
Assuming: Let \(V\) non-zero finite dimensional
\(\mathbb F\) is \(\mathbb R\)
DefType: \(\mathcal L(V_{\mathbb C})\)

Smaller0: the dimension of \(V\)(as a real vector space)
Greater0: the dimension of \(V_{\mathbb C}\) as a complex vector space
Page: 277
Chapter: 9
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
Let \(V\) be a \(\mathbb F\) vector space
Index: 4
Smaller to greater: equals
Assuming: Let \(V\) non-zero finite dimensional
\(\mathbb F\) is \(\mathbb R\)

Smaller0: \(v_1,\dots,v_n\) is a basis of \(V\)(as a real vector space)
Greater0: \(v_1,\dots,v_n\)
Page: 277
Chapter: 9
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) vector space
\(v_1,\dots,v_n\in V\) .
Index: 4
Smaller to greater: iff
Assuming: Nothing
Greater1: a basis of \(V_{\mathbb C}\) (as a complex vector space)
Connect Greater0: is

Smaller0: \(\mathcal M(T)\)
Greater0: \(\mathcal M(T_{\mathbb C})\)
Page: 278
Chapter: 9
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) vector space
\(T\in\mathcal{L}(V)\).
With basis \(v_1,\dots,v_n\in V\)
Index: 7
Smaller to greater: equals
Assuming: Let \(V\) non-zero finite dimensional
\(\mathbb F\) is \(\mathbb R\)

Smaller0: the minimal polynomial of \(T_{\mathbb C}\)
Greater0: the minimal polynomial of \(T\)
Page: 279
Chapter: 9
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
Let \(V\) be a vector space
Let \(T\in\mathcal{L}(V)\).
Index: 10
Smaller to greater: equals
Assuming: Let \(V\) non-zero finite dimensional
\(\mathbb{F=R}\)

Smaller0: \(V\) is non-zero finite dimensional
Greater0: an invariant subspace
Page: 279
Chapter: 9
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\).\(T\in\mathcal{L}(V)\).
Index: 8
Prefix Greater: \(T\) admits
Assuming: Nothing
Greater1: 1 or 2
Connect Greater0: of dimension

Smaller0: \(\lambda\) is an eigenvalue of \(T_{\mathbb C}\)
Greater0: \(\lambda\) is an eigenvalue of \(T\)
Page: 280
Chapter: 9
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
Let \(V\) be a \(\mathbb F\) vector space
Let \(T\in\mathcal{L}(V)\).
\(\lambda\in\mathbb R\)
Index: 11
Smaller to greater: iff
Assuming: Let \(V\) non-zero finite dimensional
\(\mathbb F\) is \(\mathbb R\)

Smaller0: \((T_{\mathbb C}-\lambda I)^j(u+iv)=0\)
Greater0: \((T_{\mathbb C}-\overline\lambda I)^j(u-iv)=0\)
Page: 281
Chapter: 9
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
Let \(V\) be a \(\mathbb F\) vector space
Let \(T\in\mathcal{L}(V)\).
\(\lambda\in\mathbb C\),
\(j\) is a nonnegative integer
\(u,v\in V\).
Index: 12
Smaller to greater: iff
Assuming: \(V\) non-zero finite dimensional
\(\mathbb F\) is \(\mathbb R\)

Smaller0: the multiplicity of \(\lambda\) as an eigenvalue of \(T_{\mathbb C}\)
Greater0: the multiplicity of \(\overline\lambda\) as an eigenvalue of \(T_{\mathbb C}\)
Page: 282
Chapter: 9
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\).
\(T\in\mathcal{L}(V)\).
\(\lambda\in\mathbb C\)
Index: 17
Smaller to greater: equals
Hide middle: yes
Assuming: Let \(V\) non-zero finite dimensional
\(\mathbb F\) is \(\mathbb R\)

Name: The characteristic polynomial of \(T\)
Definition: The characteristic polynomial of \(T_{\mathbb C}\).
Page: 283
Chapter: 9
Section: A
Variables: \(V\) be a \(\mathbb R\) vector space.
\(T\in\mathcal{L}(V)\).
Index: 21
Assuming: \(V\) non-zero finite dimensional

Question: The coefficients of the characteristic polynomial of \(T_{\mathbb C}\) are
Answer: real.
Page: 283
Chapter: 9
Section: A
Variables: \(V\) be a \(\mathbb R\) vector space
\(T\in\mathcal L(V)\)
Index: 20
Assuming: Let \(V\) non-zero finite dimensional

Smaller0: is normal
Greater0: with respect to every/some orthonormal basis of \(V\) has the form \(\left(\begin{array}{ll}a&-b\\b&a\end{array}\right)\) with \(b\ne 0\)
Page: 287
Chapter: 9
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) vector space
\(T\in\mathcal{L}(V)\).
Index: 27
Smaller to greater: iff
Hide middle: yes
Prefix Greater: the matrix of \(T\)
Assuming: Let \(V\) 2 dimensional
\(\mathbb F\) is \(\mathbb R\),
Smaller1: is not self-adjoint
Prefix Smaller: \(T\)
Connect Smaller0: and \(T\)

Smaller0: \(T\) is normal
Greater0: \((T_{\mid U})^*\)
Page: 288
Chapter: 9
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\)-inner product space,
\(U\) a subspace of \(V\)
\(T\in\mathcal{L}(V)\).
Index: 30c, d
Hide middle: yes
Assuming: \(V\) finite dimensional nonzero
Smaller1: \(U\) is invariant under \(T\)
Greater1: \((T^*)_{|U}\)
Connect Greater0: equals
Greater2: a normal operators
Connect Greater1: and \(T_{|U^\bot}\in\mathcal L(U^\bot)\) is

Name: The inverse of \(A\)
Definition: A square matrix \(B\) such that \(AB=BA=I\).
Page: 296
Chapter: 10
Section: B
Variables: \(A\) a square matrix
Notation: \(A^{-1}\)
Index: 2
Extra: Let \(\mathbb F\) a field

Name: An invertible matrix
Definition: A square matrix \(A\) such that there exists \(B\) such that \(AB=BA=I\).
Page: 296, 46
Chapter: 10, 2 Kenneth
Section: B,
Variables: \(\mathbb F\) a field,
Index: 2
Definition2: a matrix whose rows are linearly independant in \(\mathbb F^n\)
Name2: Non singular matrix

Name: Singular square matrix
Definition: A square matrix \(A\) such that there exists no \(B\) such that \(AB=BA=I\).
Page: 296
Chapter: 10
Section: A
Index: 2

Name: Matrix of the identity with respect to two bases
Definition: \(\mathcal M(I,(u_1,\dots,u_n),(v_1,\dots,v_n))\).
Page: 297
Chapter: 10
Section: A
Variables: \(\mathbb F\) a field
\(V\) a \(\mathbb F\)-vector space with basis \((u_1,\dots,u_n)\) and \((v_1,\dots,v_n)\).
Index: 5
Typ: Invertible matrix

Smaller0: \(\mathcal M(ST,(u_1,\dots,u_n),(w_1,\dots,w_n))\)
Greater0: \(\mathcal M(S,(u_1,\dots,u_n),(v_1,\dots,v_n))\mathcal M(T,(v_1,\dots,v_n),(w_1,\dots,w_n))\)
Page: 297
Chapter: 10
Section: A
Variables: \(\mathbb F\) a field
\(V\) be a finite-dimensional nonzero \(\mathbb F\)-vector space with basis \((u_1,\dots,u_n)\), \((v_1,\dots,v_n)\) and \((w_1,\dots,w_n)\)
\(T,S\in\mathcal{L}(V)\)
Index: 4
Smaller to greater: =
Hide middle: yes
Assuming: Nothing

Name: Invertible matrix
Page: 297
Section: A
Chapter: 10
Under: inverse

Smaller0: \(\mathcal M(I,(u_1,\dots,u_n),(v_1,\dots,v_n))\)
Greater0: \(\mathcal M(I,(v_1,\dots,v_n),(u_1,\dots,u_n))\)
Page: 297
Chapter: 10
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\) inner product space with basis \((u_1,\dots,u_n)\) and \(v_1,\dots,v_n\).
Index: 5
Smaller to greater: is the inverse of
Hide sides: yes

Smaller0: \[\mathcal M(T,(u_1,\dots,u_n))\]
Greater0: \(\mathcal M(I,(v_1,\dots,v_n),(u_1,\dots,u_n))\mathcal M(T,(v_1,\dots,v_n))\mathcal M(I,(u_1,\dots,u_n),(v_1,\dots,v_n))\)
Page: 298
Chapter: 10
Section: A
Variables: \(\mathbb F\) a field,
Let \(V\) be a \(\mathbb F\) a vector spaces,
\((u_1,\dots,u_n)\) and \((v_1,\dots,v_n)\).
Let \(T\in\mathcal{L}(V)\).
Index: 7
Smaller to greater: =
Hide middle: yes
Assuming: \(V\) non-zero finite dimensional
Connect Smaller0: .

Name: Trace of \(T\)
Definition: the sum of the generalized eigenvalues of \(T\) or of \(T_{\mathbb C}\)
Page: 299
Chapter: 10
Section: A
Variables: \(\mathbb F\) a field,
Let \(V\) be a \(\mathbb F\) inner product space with basis \(v_1,\dots,v_n\).
with basis \((u_1,\dots,u_n)\)
Let \(T\in\mathcal{L}(V)\).
\(M\in \mathbb F^{n\times n}\)
Index: 9 10.12 10.16
Kind: Example
Definition2: the negative of the coefficient of \(z^{\text{dim }V-1}\) in the characteristic polynomial of \(T\).
Definition3: the same thing, for \(\mathcal (T,(u_1,\dots,u_n))\)
Assuming: Let \(V\) non-zero finite dimensional

Name: The trace of matrix
Page: 301, 302, 303
Section: A
Chapter: 10
Index: 14, 15, 18
Under: applying \(T\mapsto P^{-1}TP\) with \(P\) invertible
Condition: invariant
Under2: changing of basis
Closure3: commutes
Under3: sum of matrices
Not4: does not
Closure4: commute
Under4: multiplication
CounterExample4: matrix 0 and I

Name: The trace of \(M\)
Definition: The sum of the diagonal entries of \(M\).
Page: 302
Chapter: 10
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(M\) be a square matrix
Index: 15

Question: \(ST-TS\) is
Answer: not the identity
Page: 304
Chapter: 10
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
Let \(V\) be a \(\mathbb F\) inner product space.
Let \(S,T\in\mathcal L(V)\).
Index: 19
Assuming: Let \(V\) non-zero finite dimensional

Name: The deteminant of \(T\)
Definition: The product of the generalized eigenvalues of \(T\) (or of \(T_{\mathbb C}\) if \(V\) is a real-vector space).
Page: 307, 308
Chapter: 10
Section: B
Variables: \(\mathbb F\) a field
\(V\) be a \(\mathbb F\) a vector space
\(T\in\mathcal{L}(V)\).
Index: 20 22
Definition2: \((-1)^{\text{dim }V}\) times the constant term of the characteristic polynomial of \(T\).
Definition3: the same thing, for \(\mathcal M(T,(v_1,\dots,v_n))\)
Abbreviation: \(\text{det }T\)
Assuming: Let \(V\) non-zero finite dimensional

Name: A permutation of \((1,\dots,n)\)
Definition: A list \((m_1,\dots,m_n)\) that contains each of the numbers \(1,\dots,n\) exactly once.
Page: 311
Chapter: 10
Section: B
Index: 27
Kind: Definition

Name: The set of permutations of \((1,\dots,n)\).
Page: 311
Chapter: 10
Section: B
Index: 27
Kind: Definition
Abbreviation: \(\text{perm }n\)

Name: The sign of \(p\)
Definition: 1 if the numbers of pairs of integers \((j,k)\) with \(1\le j<k\le n\) such that \(j\) appears after \(k\) in the list \((m_1,\dots,m_n)\) is even and -1 if the number of such pairs is odd.
Page: 312
Chapter: 10
Section: B
Variables: \(p=(m_1,\dots,m_n)\) a permutation.
Index: 30
Kind: Definition

Smaller0: \(p'\) is \(p\) with two entries interchanged in a permutation
Greater0: the negation of the sign of permutation of \(p\)
Page: 313
Chapter: 10
Section: B
Variables: \(p,p'\) two permutations.
Index: 32
Hide middle: yes
Prefix Greater: the sign of the permutation of \(p'\) is

Smaller0: the matrix obtained from \(A\) by interchanging two columns
Greater0: \(-\text{det }B\)
Page: 315
Chapter: 10
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
Let \(A,B\) be \(n\)-by-\(n\) matrices.
Index: 36
Hide middle: yes
Prefix Greater: \(\text{det }A=\)
Assuming: Nothing
Prefix Smaller: \(B\) is

Smaller0: \(A\) has two equal columns
Greater0: \(0\)
Page: 316
Chapter: 10
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
\(A\) be an \(n\)-by-\(n\) matrix
Index: 36
Hide middle: yes
Prefix Greater: \(\text{det }A=\)
Assuming: \(A\) is an \(n\)-by-\(n\) matrix.

Question: \(\text{det}(ST)\)
Answer: equals \((\text{det}T)(\text{det}S)\)
Page: 320
Chapter: 10
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\)-vector space.
\(S,T\in\mathcal L(V)\).
Index: 44
Assuming: \(V\) finite dimensionial non-zero

Question: According to most applied mathematicians, determinants should be used in serious numeric calculation
Answer: rarely
Page: 320
Chapter: 10
Section: B
Index: 44

Smaller0: \(T\) is an isometry
Greater0: \(=1\)
Page: 321
Chapter: 10
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\)-vector space.
\(T\in\mathcal L(V)\).
Index: 45
Prefix Greater: \(|\text{det} T|\)
Assuming: \(V\) finite dimensionial non-zero

Smaller0: [$]\abs{\text{det }T}[/$]
Greater0: \(\text{det }\sqrt{T^*T} \)
Page: 323
Chapter: 10
Section: B
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\).
\(V\) be a \(\mathbb F\)-vector space.
\(T\in\mathcal L(V)\).
Index: 47
Smaller to greater: is equal to
Hide middle: yes
Assuming: \(V\) finite dimensionial non-zero

Smaller0: \(T\) is an isometry
Greater0: \(=\text{volume }\Omega\)
Page: 325
Chapter: 10
Section: B
Variables: \(n\in\mathbb N\).
\(\Omega\subseteq \mathbb R^n\).
\(T\in\mathcal L(\mathbb R^n)\)
Index: 53
Hide middle: yes
Prefix Greater: \(\text{volume }T(\Omega)\)

Smaller0: \[\text{volume }T(\Omega)\]
Greater0: \[|\text{det }T|(\text{volume }\Omega)\]
Page: 326
Chapter: 10
Section: B
Variables: \(n\in\mathbb N\).
\(\Omega\subseteq \mathbb R^n\).
\(T\in\mathcal L(\mathbb R^n)\)
Index: 54
Kind: Definition
Smaller to greater: =

Name: Derivative of \(\sigma\) at \(x\)
Definition: The unique operator \(T\in\mathcal L(\mathbb R^n)\) satisfying
\[\lim_{y\to0}\frac{\parallel \sigma(x+y)-\sigma(x)-Ty\parallel }{\parallel y\parallel }=0.\]
Page: 327
Chapter: 10
Section: B
Variables: \(n\in\mathbb N\).
\(\Omega\subseteq \mathbb R^n\) open,
\(\sigma:\Omega\to\mathbb R^n\)
Notation: \(\sigma'(x)\)
Index: 56
Kind: Definition
Assuming: Nothing

Name: A function differentiable at \(x\)
Definition: A function \(\sigma\) such that there exists an operator \(T\in\mathcal L(\mathbb R^n)\) such that
[$$]
\lim_{\mathbb R^n\ni y\to0}\frac{\norm{\sigma(x+y)-\sigma(x)-Ty}}{\norm{y}}=0.
[/$$]
Page: 327
Chapter: 10
Section: B
Variables: \(n\in\mathbb N\),
\(\Omega\subseteq\mathbb R^n\)
Index: 56
Kind: Definition

Smaller0: [$]\int_{\sigma(\Omega)}f(y)\dd y[/$]
Greater0: [$$]\int_\Omega f(\sigma(x))|\text{det }\sigma'(x)|\dd x[/$$]
Page: 328
Chapter: 10
Section: B
Variables: \(n\in\mathbb N\)
\(\Omega\subseteq \mathbb R^n\) open,
\(\sigma:\Omega\to\mathbb R^n\)
Index: 58
Smaller to greater: =
Hide middle: yes
Assuming: \(\sigma\) is differentiable on \(\Omega\)

Name: The partial derivative with respect to the \(j\)th coordinate of \(\sigma\)
Definition: A function \(\sigma\) such that there exists an operator \(T\in\mathcal L(\mathbb R^n)\) such that
[$$]
\lim_{\mathbb R\ni y\to0}\frac{\norm{\sigma(x+y\cdot e_j)-\sigma(x)-Ty}}{\abs{y}}=0.
[/$$]
Page: 328
Chapter: 10
Section: B
Variables: \(\Omega\subseteq\mathbb R^n\)
\(\sigma:\Omega\to\mathbb R^n\)
\(\sigma:\Omega\to\mathbb R^n\)
Notation: \(D_j\sigma\)
Index: 56
Kind: Definition

Question: The matrix of derivative of \(\sigma\) at \(x\) equals
Answer: \[\left(\begin{array}{lll}D_1\sigma_1(x)&\dots& D_n\sigma_n (x)\\\dots&&\dots\\D_1\sigma_n(x)&\dots&D_n\sigma_n(x)\end{array}\right).\]
Page: 328
Chapter: 10
Section: B
Variables: \(n\in\mathbb N\).
\(\Omega\subseteq \mathbb R^n\) open,
\(\sigma:\Omega\to\mathbb R^n\),
\(x\in\Omega\).
Index: 57
Assuming: \(\sigma\) differentiable at \(x\)

Smaller0: [$$]\int_{-1}^1\int_{-\sqrt{1-x^2} }^{\sqrt{1-x^2} }f(x,y)\dd y\dd x[/$$]
Greater0: [$$]\int_0^{2\pi}\int_0^1f(r\cos\theta,r\sin \theta)r \dd r\dd\theta[/$$]
Page: 329
Chapter: 10
Section: A
Index: 59
Kind: Example
Smaller to greater: =
Hide middle: yes

Smaller0: [$$]\int_{-1}^1\int_{-\sqrt{1-x^2} }^{\sqrt{1-x^2} }\int_{-\sqrt{1-x^2-y^2} }^{\sqrt{1-x^2-y^2} }f(x,y,z)\dd y\dd x\dd z[/$$]
Greater0: [$$]\int_0^{2\pi}\int_0^{\pi}\int_0^1f(\rho\sin\phi\cos\theta,\rho\sin\phi\sin\theta,\rho\cos\phi)\rho^2 \sin\phi\dd \rho \dd \phi d\theta[/$$]
Page: 330
Chapter: 10
Section: B
Index: 60
Smaller to greater: is equal to
Hide middle: yes

Page: 234
Chapter: 7
Section: 2
Name: The set of polynomials with coefficients in \(R\)
Set: The set of terms of the form \(\sum_{i=0}^n a_ix^i\)
Variables: \(R\) a ring with identity.
Notation: \(R[x]\), or \(\mathcal P(F)\) (in linear algebra done right)
*: \(\left(\sum_{i=0}^n a_ix^i,\sum_{i=0}^m b_ix^i\right)\mapsto\sum_{i=0}^{n+m}\left(\sum_{j=0}^ia_jb_{i-j}\right)x^i\)
+: \(\left(\sum_{i=0}^n a_ix^i,\sum_{i=0}^m b_ix^i\right)\mapsto\sum_{i=0}^n(a_i+b_i)x^i+\sum_{i=n+1}^mb_ix^i\)
0: Empty sum
1: \(1 x^0\)
-: \(\sum_{i=0}^n a_ix^i\to\sum_{i=0}^n -a_ix^i\)

Name: Linear map over \(F[x]\)
Page: 053
Chapter: 3
Section: A
Variables: \(\mathbb F\) be \(\mathbb R\) or \(\mathbb C\)
Index: 4
Kind: Example
Example: Differentiation
Example2: integration
Example3: multiplication by a polynomial

Name: A scalar
Definition: An element of \(\mathbb F\)
Page: 4
Chapter: 1
Section: A
Variables: \(\mathbb F\) a field
Kind: Cor

Name: A vector
Definition: An element of \(V\)
Page: 13
Chapter: 1
Section: A
Variables: \(\mathbb F\) a field
\(V\) a vector space
Index: 20
Kind: Cor
Name2: A point

Smaller0: \(p\) only has real coefficients
Greater0: closed under conjugation
Page: 126
Chapter: 4
Section: A
Variables: \(p\in\mathcal P(\mathbb C)\)
\(\lambda\) be a zero of \(p\).
Index: 15
Prefix Greater: the zeros of \(p\) are