Title: Theory / Important / Math / Al gebra / Linear / Kenneth Kunze+8
Tags: hasSeenConjDef toBinaryRelation examplesSwitchExample1 closed counterExample fromAlgorithm fromMath hasNotNewEffectReturnsMeaning

Name: A commutative operation
Definition: An operation \(+\) such that \(a+b=b+a\) for each values \(a\) and \(b\)
Page: 1

Name: An associative operation
Definition: An operation \(+\) such that \((a+b)+c=a+(b+c)\) for each values \(a, b\) and \(c\)
Page: 1

Name: A zero for \(+\)
Definition: An element such that \(0+a=a+0=a\) for each \(a\)
Page: 1
Variables: \(+\) a binary operation
Index: 3
Cardinal: At most one

Name: An inverse of \(a\) for \(+\)
Definition: An element \(b\) such that \(a+b=b+a=0\)
Page: 1
Variables: \(+\) a binary operation, with 0
\(a\) in the domain
Notation: \(-a\)
Index: 4
Cardinal: At most one

Name: \(\cdot\) is distributive over \(+\)
Definition: \(a(b+c)=ab+ac\) for each \(a,b,c\).
Page: 2
Chapter: 1
Variables: \(+,\cdot\) binary operations
Index: 9

Name: A field
Definition: \((F,+)\) is an Abelian group
Page: 1, 2
Chapter: 1
Definition2: \((F,\cdot)\) is an Abelian group
Example: \(\mathbb Q\), \(\mathbb R\), \(\mathbb C\)
Conjdef: A tuple \((F,+,\cdot)\) such that
Definition3: \(\cdot\) is distributive over \(+\)
Denoted by: \(F\)

Name: A group
Definition: \(+\) admits a 0 in \(F\)
Page: 1, 2
Chapter: 1
Definition2: \(+\) is associative
Conjdef: A tuple \((F,+)\) such that
Definition3: each elements of \(F\) admits an inverse

Name: A subfield of \(F\)
Definition: A subset of \(F\) which is a field when \(+\) and \(\cdot\) are restricted to this subset
Page: 2
Chapter: 1
Variables: \((F,+,\cdot)\) a field

Name: The characteristic of \(F\)
Definition: The least \(n\in\mathbb N\) such that \(n\cdot 1_F=0_F\) if it is finite, 0 otherwise.
Page: 3
Chapter: 1
Variables: \(F\) a field

Definition: \(0_F\) if \(n=0\)
\((n-1)x+_Fx\) if \(n{\gt}0\)
\(-_F((-n)x)\) if \(n{\lt}0\)
Page: 3
Chapter: 1
Variables: \(F\) a field,
\(x\in F\),
\(n\in\mathbb Z\)
Notation: \(nx\)

Name: A system of \(m\) linear equation in \(n\) unknowns
Definition: A system \((\sum A_{i,j}x_j=y_i)_{1\le i\le n}\), with \(A_{i,j}\) and \(y_i\) fixed
Page: 3
Chapter: 1
Section: 1.2

Name: A solution of the system
Definition: An \(n\)-tuple of values such that the equation holds
Page: 4
Chapter: 1
Section: 1.2
Variables: \((\sum A_{i,j}x_j=y_i)_{1\le i\le n}\) a system of \(m\) linear equation in \(n\) unknowns

Name: An homogeneous system of equations
Definition: A system of the form \((\sum A_{i,j}x_j=0)_{1\le i\le n}\)
Page: 4
Chapter: 1
Section: 1.2

Name: A linear combination of the equations
Definition: \(\sum_j\left(\sum_i c_i A_{i,j}\right)x_j=\sum_i c_iy_i\)
Page: 4
Chapter: 1
Section: 1.2
Variables: \((\sum A_{i,j}x_j=y_i)_{1\le i\le n}\) a system of \(m\) linear equation in \(n\) unknowns
Definition2: \(\sum_ic_i \left(\sum_i A_{i,j}x_j\right)=\sum_i c_iy_i\)
Conjdef: An equation of the form

Page: 4,5
Definition: each equations of a system is a linear combination of equations of the other system
Chapter: 1
Section: 1.2
Name: Equivalent system of linear equations
Kind: Theo
Definition2: the systems of linear equations with the same solutions
Symmetry: The relation on systems of linear equations such that

Name: The matrix of coefficients of the system
Definition: \(\left[\begin{array}{lll}A_{11}&\dots&A_{1n}\\\vdots &&\vdots\\A_{m1}&\dots& A_{mn}\end{array}\right]\)
Page: 6
Chapter: 1
Section: 1.3
Variables: \((\sum A_{i,j}x_j=y_i)_{1\le i\le n}\) a system of \(m\) linear equation in \(n\) unknowns

Name: An \(m\times n\) matrix over \(F\)
Definition: A function from \([1,m]\times[1,n]\) to \(F\)
Page: 6
Chapter: 1
Section: 1.3
Variables: \(F\) a field

Name: Elementary row operations on an \(m\times n\) matrix
Definition: Multiplication of one row by a non-zero scalar
Page: 6, 7
Chapter: 1
Section: 1.3
Variables: \(F\) a field
Index: 2
Kind: Theo
Definition2: Replacement of a row by this row plus \(c\) times another row
Conjdef: All of
Definition3: interchange of two rows
Property: Closed under inverse

Page: 7
Definition: The equivalence relation containing a matrix, and the result of applying a row operation to this matrix
Chapter: 1
Section: 1.3
Variables: \(F\) a field
Name: Row equivalence
Kind: Def
Context: Matrix

Smaller0: \(A\) and \(B\) are row equivalent
Greater0: \(A\) and \(B\) have the same solutions
Page: 7
Chapter: 1
Section: 1.3
Variables: \(F\) a field,
\(A,B\) be \(m\times n\) matrixes
Kind: Def
Hide middle: yes
Context: Matrix

Name: A row-reduced matrix
Definition: the first non-zero entry in each non-zero row is equal to 1
Page: 9
Chapter: 1
Section: 1.3
Variables: \(F\) a field
Kind: Def
Definition2: each column containing the leading non-zero entry of some row has all its other entries 0
Example: Identity matrix
Conjdef: A matrix such that
Example2: 0 matrix
Example3: \(\left[\begin{array}{lllll}0&1&-3&0&\frac 12\\&&&1&2\\&&&&\end{array}\right]\)

Name: Kronecker delta
Definition: 1 if \(i=j\), 0 otherwise
Page: 9
Chapter: 1
Section: 1.3
Variables: \(F\) a field,
\(i,j\) some values
Notation: \(\delta_{i,j}\)
Kind: Def

Page: 10, 11
Implementation: For \(i\) in the set of row:
if the \(i\)-th row is 0, pass
otherwise, let \(j\) the first coefficient such that the \(ij\)-th entry is nonzero. Divide the row by \(A_{ij}\)
Add this row, divided by \(A_{i'j}\), to each row \(i'\)
Section: 1.3
Chapter: 1
Effect: A row-reduced matrix row-equivalent to \(A\),
Variables: \(F\) a field,
\(A\) a \(m\times n\) matrix
Index: 5
Kind: Def, Theo

Question: The rows of \(AB\) are linear combinations of
Answer: the rows of \(B\)
Page: 17
Chapter: 1
Section: 5
Variables: \(A\in\mathbb F^{m\times n}\)
\(B\in\mathbb F^{n\times p}\)

Name: An elementary matrix
Definition: A matrix which can be obtained from the \(m\times m\) identity matrix by means of a single elementary row operation
Page: 20
Chapter: 1
Section: 5
Kind: Def
Example: for \(m=2\), it is
\(\left[\begin{array}{ll}&1\\1&\end{array}\right]\)
\(\left[\begin{array}{ll}c&1\\1&\end{array}\right]\)

Name: An elementary matrix
Definition: A matrix which can be obtained from the \(m\times m\) identity matrix by means of a single elementary row operation
Page: 20, 22
Chapter: 1
Section: 5
Context: duplicate
Index: , 11
Kind: Def, Theo
Example: for \(m=2\), it is
\(\left[\begin{array}{ll}&1\\1&\end{array}\right]\)
\(\left[\begin{array}{ll}1&c\\&1\end{array}\right]\)
\(\left[\begin{array}{ll}1&\\c&1\end{array}\right]\)
\(\left[\begin{array}{ll}d&\\&1\end{array}\right]\)
\(\left[\begin{array}{ll}1&\\&d\end{array}\right]\) with \(d\ne0\)
Typ: Invertible

Smaller0: \(e(A)\)
Greater0: \(e(I_m)A\)
Page: 20
Chapter: 1
Section: 5
Variables: \(e\) an elementary row operation
\(A\in\mathbb F^{m\times n}\)
Index: 9
Kind: Theo
Smaller to greater: =

Page: 7, 20, 23
Definition: \(B\) can be obtained from \(A\) by a finite sequence of elementary row operations
Chapter: 1
Section: 3, 5
Name: Row-equivalent matrices
Kind: Def, cor
Definition2: \(B=\prod e_i(I_{m})A\), with \(e_i\) being elementary row operations
Definition3: \(B=PA\) with \(P\) invertible
Set: \(m\times n\)-matrices \(A,B\)

Name: A left-inverse of \(A\)
Definition: A matrix \(B\in\mathbb F^{m\times n}\) such that \(BA=I_n\)
Page: 22
Chapter: 1
Section: 6
Variables: \(A\in\mathbb F^{m\times n}\)
Kind: Def

Name: A right-inverse of \(A\)
Definition: A matrix \(B\in\mathbb F^{m\times n}\) such that \(AB=I_m\)
Page: 22
Chapter: 1
Section: 6
Variables: \(A\in\mathbb F^{m\times n}\)
Kind: Def

Smaller0: \(A\) has right and left inverse
Greater0: \(A\)'s right and left inverse are equal
Page: 22
Chapter: 1
Section: 6
Variables: \(A\in\mathbb F^{m\times n}\)
Kind: Def
Smaller to greater: iff

Smaller0: \(e_0\dots e_{m-1}A=I_n\)
Greater0: \(e_0\dots e_{m-1}I_n=A^{-1}\)
Page: 22
Chapter: 1
Section: 6
Variables: \(A\in\mathbb F^{n\times n}\),
\((e_i)_{i<m}\) a sequence of row operations
Kind: Cor
Smaller to greater: iff

Smaller0: \(m=n\)
Greater0: \(A\) is invertible
Page: 24
Chapter: 1
Section: 6
Variables: \(A\in\mathbb F^{n\times m}\),
Kind: Cor
Smaller to greater: iff
Smaller1: \(A\) has a left or right inverse

Smaller0: each \(A_i\) is invertible
Greater0: invertible
Page: 24
Chapter: 1
Section: 6
Variables: \(A_1,\dots,A_k\in\mathbb F^{n\times n}\),
Kind: Cor
Smaller to greater: iff
Prefix Greater: \(\prod A_i\) is

Question: How to transform result about row equivalent to column equivalence ?
Answer: Multiply the elementary/transformation matrices on the right instead of on the left
Page: 26
Chapter: 1
Section: 6
Variables: \(A_1,\dots,A_k\in\mathbb F^{n\times m}\),
Kind: Cor

Question: The dimension of \(W_1+W_2\) is
Answer: \(\mathrm{dim}(W_1)+\mathrm{dim}(W_2)-\mathrm{dim}(W_1\cap W_2)\)
Page: 46
Chapter: 2
Variables: \(F\) a field
\(V\) a \(F\)-vector space
\(W_1,W_2\subseteq V\) subspaces
Index: 6
Kind: Theo

Smaller0: \(\{v,w\}\) is a dependend set of vectors
Greater0: \(v\) is a multiple of \(w\) or reciprocally
Page: 48
Chapter: 2
Variables: \(F\) a field
\(V\) a \(F\)-vector space
\(v,w\in V\)
Index: 1
Kind: Ex
Smaller to greater: iff

Name: An ordered basis of \(V\)
Definition: A finite sequence of vectors which is linearly independent and spans \(V\)
Page: 50
Chapter: 2
Section: 4
Variables: \(F\) a field
\(V\) a \(F\)-vector space
Index: 1
Kind: Ex
Denoted by: \(\mathscr B\)

Name: The \(i\)th coordinate of \(\alpha\) relative to \(\mathscr B\)
Definition: the unique number \(x_i\) such that \(\alpha=\sum_{j=1}^n x_j\alpha_j\) for some \(x_j\in F\)
Page: 50
Chapter: 2
Section: 4
Variables: \(F\) a field
\(V\) a \(F\)-vector space
\(\mathscr B=\{\alpha_1,\dots,\alpha_n\}\) an ordered basis
\(1\le i\le n\)
\(\alpha\in V\)
Index: 1
Kind: Ex

Question: The \(i\)th coordinate of \(\alpha+\beta\) relative to \(\mathscr B\)
Answer: is the sum of the \(i\)th coordinate of \(\alpha\) and of \(\beta\) relative to \(\mathscr B\)
Page: 50
Chapter: 2
Section: 4
Variables: \(F\) a field
\(V\) a \(F\)-vector space
\(\mathscr B=\{\alpha_1,\dots,\alpha_n\}\) an ordered basis
\(1\le i\le n\)
\(\alpha,\beta\in V\)
Index: 1
Kind: Ex

Definition: The unique tuple \((x_1,\dots,x_n)\) such that \(\alpha=\sum x_i\alpha_i\)
Page: 52
Chapter: 2
Section: 4
Variables: \(F\) a field
\(V\) a \(F\)-vector space
\(\mathscr B=\{\alpha_1,\dots,\alpha_n\}\) an ordered basis
\(\alpha\in V\)
Notation: \([\alpha]_{\mathscr B}\)
Index: 7
Kind: Theo
Definition2: \((M_{1,j},\dots,M_{n,j})=[\alpha'_j]_{\mathscr{B}}\)

Name: The transition matrix from \(\mathscr B\) to \(\mathscr B'\)
Definition: \([\alpha]_{\mathscr B}=M[\alpha]_{\mathscr B'}\) and \([\alpha]_{\mathscr B'}=M^{-1}[\alpha]_{\mathscr B}\) for each \(\alpha\in V\)
Page: 50
Chapter: 2
Section: 4
Variables: \(F\) a field
\(V\) a \(F\)-vector space
\(\mathscr B=\{\alpha_1,\dots,\alpha_n\}\), \(\mathscr B'=\{\alpha_1',\dots,\alpha_n'\}\) ordered basis
Index: 1
Kind: Ex
Conjdef: The unique matrix \(M\) such that, equivalently
Denoted by: \(P\)
Property: Invertible

Name: The set of basis of \(V\)
Page: 53
Section: 4
Chapter: 2
Index: 8
Variables: \(F\) a field
\(V\) a \(F\)-vector space
Kind: Theo
Under: multiplication by a matrices
Condition: invertible

Smaller0: \(P\) is invertible
Greater0: a basis of \(V\)
Page: 53
Chapter: 2
Section: 4
Variables: \(F\) a field
\(V\) a \(F\)-vector space
\(\mathscr B=\{\alpha_1,\dots,\alpha_n\}\) an ordered basis
\(P\in F^{n\times n}\)
Index: 7, 8
Kind: Theo
Smaller to greater: iff
Prefix Greater: \(\{P\alpha_1,\dots,P\alpha_n\}\) is