Title: Theory / Important / Math / Al gebra / Linear / Advanced - Brown/
Tags: fromMath mathToDef mathToSee automaticImplies examplesSwitchExample1 questionExample

Definition: The set of vertical vectors \(a_1,a_2,\dots,\) with \(a_i\in F\) and \(a_i=0\) for all but finitely many \(i\)
Page: 6
Chapter: 2
Section: 1
Variables: \(F\) a fieald
Notation: \(F^\infty\)

Definition: The set of vertical vectors \(a_1,a_2,\dots,\) with \(a_i\in F\)
Page: 6
Chapter: 2
Section: 1
Variables: \(F\) a fieald
Notation: \(F^{\mathbb N}\)
Index: 7
Kind: Ex

Name: The \(n\) dimensional sphere
Definition: The set of \(n\)-tuple whose sum of square is 1.
Page: 6
Chapter: 2
Section: 1
Variables: \(n\in\mathbb N\)
Notation: \(S^{n-1}\)
Index: 8
Kind: Ex

Smaller0: The tangent plane of \(S^{n-1}\) at \((a_1,\dots,a_n)\)
Greater0: \(\sum 2a_i(x_i-a_i)\)
Page: 6
Chapter: 2
Section: 1
Variables: \(n\in\mathbb N\)
\(p\in S^{n-1}\)
Index: 8
Kind: Ex
Smaller to greater: is equal to
Hide middle: yes

Name: An \(F\)-algebra
Definition: A ring \((A,+,\times,1)\) and \(f:F\to A\) sending \(1_F\) to \(1_A\), such that for all \(a\in A,c\in F \), \(af(c)=f(c)a\).
Page: 11
Chapter: 2
Section: 1
Variables: \(F\) a field
Index: 26
Kind: DefEx
Definition2: A \(F\)-vector space with a bilinear product
Example: \(\mathbb C\) and the quaternions over \(\mathbb R\)
Example2: The ring of square matrices
Example3: \(\mathbb R^3\) with cross-product

Name: Zorn's lemma
Definition: Let \(X\) be any partially ordered set with the property that every chain in \(X\) has an upper bound. Then \(X\) contains at least one maximal element.
Page: 14
Chapter: 2
Section: 1
Index: 4
Kind: Theo

Name: Infinite countable dimensional \(F\)-vector space
Page: 17
Chapter: 2
Variables: \(F\) a field
Index: 19
Kind: Ex
Example: \(F[x]\)

Name: Vector space without countable basis
Page: 18
Chapter: 2
Index: 20
Kind: Ex
Example: \(\mathbb R^{\mathbb N}\)

Q: \(V\) is [longer name of the basis, in case of need for disambiguation] if [...].
A: a Hamel basis; each element of \(V\) is a linear combination of elements of \(B\)
Page: 18
Chapter: 2
Variables: \(V\) a vector space and \(B\subseteq V\).
Index: 20
Kind: Ex

Smaller0: a basis of \(W\)
Greater0: \(T\) is an isomorphism
Page: 19
Chapter: 2
Variables: \(V,W\) vector spaces,
\(T:V\to W\) and
\(B\) a basis of \(V\).
Index: 23
Kind: Cor
Smaller to greater: iff
Hide middle: yes
Prefix Smaller: \(T(B)\) is

Name: A complement of \(U\)
Definition: A subspace \(W\subseteq V\) such that \(U\oplus W=V\).
Page: 24
Chapter: 2
Section: 3
Variables: \(V\) a vector spaces and
\(U\subseteq V\) a subspace.
Index: 8
Kind: Def

Name: The (algebraic) dual space of \(V\)
Definition: \(\text{Hom}_F(V,F)\).
Page: 27
Chapter: 2
Section: 4
Variables: \(V\) a vector spaces
Notation: \(V^\lor\)
Index: 1
Kind: Def

Question: \(V\) is isomorphic to
Answer: a subspace of \(V^\lor\).
Page: 27
Chapter: 2
Section: 4
Variables: \(V\) a vector spaces
Kind: Theo

Question: The canonical injective linear map from \(V\) to \(\left(V^\lor\right)^\lor\) is
Answer: \(v\mapsto\left(\phi\mapsto\phi(v)\right)\). That is
the function which associates to \(v\) the function which associate to \(\phi\in V^\lor\) the scalar \(\phi(v)\).
Page: 29
Chapter: 2
Section: 4
Variables: \(F\) a field and \(V\) a \(F\)-vector spaces
Index: 5
Kind: Prop

Q: [...], it is [...].
A: [$]\defFun{W^\lor}{V^\lor}{\phi}{\phi\circ T}[/$]; linear
Page: 30
Chapter: 2
Section: 4
Variables: \(F\) a field and \(V,W\),
\(F\)-vector spaces and
\(T\in\text{Hom}_F(V,W)\)
Index: 6
Kind: Prop
Notation: \(T^\lor\)

Name: A left \(R\)-module
Definition: An abelian group \((G,+)\) and a map \(R\times G\to G\):
-\((r_1+r_2)m=r_1m+r_2m\),
-\(r(m_1+m_2)=rm_1+rm_2\),
-\((r_1r_2)m=r_1(r_2m)\),
-\(1_Rm=m\)
Page: 30
Chapter: 2
Section: 5
Variables: \(R\) a ring.
Index: 1
Kind: Def

Q: \([...]=[...]\) where [...].
A: [v]_{\mathcal B}; \left(\begin{array}{l}a_1\\\dots\\a_n\end{array}\right); \(v=\sum_{i=1}^na_iv_i\)
Page: 40
Chapter: 3
Section: 1
Variables: \(F\) a field, \(V\) a \(F\)-field, \(\mathcal B=\{v_1,\dots,v_n\}\) a basis of \(V\) and \(v\in V\).

Definition: \(\left(\begin{array}{lll}a_{1,1}&\dots{}&a_{1,n}\\\dots&\dots&\dots\\a_{m,1}&\dots&a_{m,n}\end{array}\right)=\left([T(v_1)]_{\mathcal{C} }|\dots|[T(v_n)]_{\mathcal{C} }\right)\) such that \(T(v_j)=\sum_{i=1}^ma_{i,j}w_i\).
Page: 41
Chapter: 3
Section: 1
Variables: \(F\) a field,
\(V,W\) \(F\)-fields,
\(\mathcal B=\{v_1,\dots,v_n\}\),
\(\mathcal C=\{w_1,\dots,w_m\}\) basis of \(V\)
\(W\) and \(T: V\to W\).
Notation: \([T]_{\mathcal B}^{\mathcal C} \)

Question: The change of basis from \(\mathcal B\) to \(\mathcal C\) is
Answer: \([\text{id}]_{\mathcal B}^{\mathcal C}\)
Page: 43
Chapter: 3
Section: 1
Variables: \(F\) a field
\(V\) a \(F\)-field
\(\mathcal B=\{v_1,\dots,v_n\},\mathcal C=\{w_1,\dots,w_m\}\) basis of \(V\).

Question: On \(T^n\), transpose and inverse
Answer: commutes.
Page: 47
Chapter: 3
Section: 1
Variables: \(n\in\mathbb N\)
\(F\) a field
Index: 6
Kind: Lem

Name: The \(T\)-annihilator of \(v\)
Definition: The unique monic nonzero polynomial \(P\) of lowest degree so that \(P(T)(v)=0\).
Page: 54
Chapter: 4
Section: 1
Variables: \(n\in\mathbb N\),
\(F\) a field,
\(V\) a \(n\)-dimensional \(F\)-vector space,
\(v\in V\) and
\(T\in\text{Hom}_F(V,V) \)
Notation: \(m_{T,v}(x)\)
Index: 3
Kind: Def

Smaller0: \(P(T)(v)=0\)
Greater0: \(m_{T,v}(x)|P(x)\)
Page: 54
Chapter: 4
Section: 1
Variables: \(n\in\mathbb N\), \(F\) a field, \(V\) a \(n\)-dimensional \(F\)-vector space, \(x\in V\) and \(T\in\text{Hom}_F(V,V) \). Let \(P\in F[x]\)
Kind: Cor

Name: The minimal polynomial of \(T\)
Definition: The unique monic polynomial \(P\) of lowest degree such that \(P(T)=0\)
Page: 54
Chapter: 4
Section: 1
Variables: \(n\in\mathbb N\),
\(F\) a field,
\(V\) a \(n\)-dimensional \(F\)-vector space,
\(x\in V\)
\(T\in\text{Hom}_F(V,V) \)
\(f\in F[x]\)
Notation: \(m_T(x)\)
Index: 10
Kind: Def

Question: \(m_T(x)\)
Answer: is equal to some \(m_{T,v}(x)\) for some \(v\in V\).
Page: 58
Chapter: 4
Section: 1
Variables: \(n\in\mathbb N\),
\(F\) a field,
\(V\) a \(n\)-dimensiona \(F\)-vector space, and
\(T\in\text{Hom}_F(V,V)\).
Index: 14
Kind: Theo

Question: The degree of \(m_T(x)\) is
Answer: at most \(n\).
Page: 58
Chapter: 4
Section: 1
Variables: \(n\in\mathbb N\)
\(F\) a field,
\(V\) a \(n\)-dimensional \(F\)-vector space
\(T\in\text{Hom}_F(V,V) \).
Index: 15
Kind: Cor

Name: The characteristic polynomial of \(A\)
Definition: \(\text{det}(x1_n-A)\)
Page: 58
Chapter: 4
Section: 1
Variables: \(n\in\mathbb N\)
\(F\) a field
\(V\) a \(n\)-dimensional \(F\)-vector space
\(A\in\text{Mat}_n(F) \)
Notation: \(c_A(x)\)
Index: 16
Kind: Def
DefType: A \(F\)-polynomial

Q: Equivalently:
-[...]
-[...].
A: \(A\) and \(PAP^{-1}\) for some \(P\in\text{GL}_n(F)\) and \(A\in\text{Mat}_n(F)\); \([T]_\mathcal B\) and \([T]_{\mathcal {C} }\) for some bases \(\mathcal{B,C}\) of \(V\) and \(T\in\mathcal L(V)\)
Page: 48
Chapter: 3
Variables: \(n\in\mathbb N\)
\(F\) a field,
\(V\) a \(n\)-dimensional \(F\)-vector space.
Index: d
Name: Similar matrices

Name: The characteristic polynomial of \(T\)
Definition: \(c_{[T]_\mathcal B}(x)\) for some basis \(\mathcal B\) of \(V\).
Page: 59
Chapter: 4
Section: 1
Variables: \(n\in\mathbb N\)
\(F\) a field
\(V\) a \(n\)-dimensional \(F\)-vector space
\(T\in\mathcal{H}om_F(V,V)\)
Notation: \(c_T(x)\)

Name: The companion matrix of \(f\)
Definition: \[\left(\begin{array}{llllll}-a_{n-1}&1&0&0&\dots&0\\-a_{n-2}&0&1&0&\dots&0\\\dots&\dots&\dots&\dots&\dots&\dots\\-a_{1}&0&0&0&\dots&1\\-a_{0}&0&0&0&\dots&0\\\end{array}\right)\]
Page: 59
Chapter: 4
Section: 1
Variables: \(n\in\mathbb N\),
\(F\) a field,
\(f(x)=x^n+\sum_{i=0}^{n-1}a_ix^i\).
Notation: \(C(f(x))\)
Index: 18
Kind: Def

Question: \(c_{C(f(x))}=\)
Answer: \(f(x)\)
Page: 59
Chapter: 4
Section: 1
Variables: \(n\in\mathbb N\)
\(F\) a field
\(V\) a \(n\)-dimensional \(F\)-vector space
\(f(x)=x^n\) a monic polynomial
Index: 19
Kind: Lem

Question: \(m_{C(f(x))}=\)
Answer: \(f(x)\)
Page: 60
Chapter: 4
Section: 1
Variables: \(n\in\mathbb N\), \(F\) a field, \(V\) a \(n\)-dimensional \(F\)-vector space, and \(f(x)=x^n+\sum_{i=0}^{n-1}a_ix^i\).
Index: 20
Kind: Theo

Question: \(m_{C(f(x)),e_n}=\)
Answer: \(f(x)\)
Page: 60
Chapter: 4
Section: 1
Variables: \(n\in\mathbb N\),
\(F\) a field,
\(V\) a \(n\)-dimensional \(F\)-vector space,
\(f(x)=x^n+\sum_{i=0}^{n-1}a_ix^i\).
Index: 20
Kind: Theo

Question: \(\{g(T_{C(f(x))})(e_n)\mid g(x)\in F[x]\}=\)
Answer: \(F^n\).
Page: 60
Chapter: 4
Section: 1
Variables: \(n\in\mathbb N\), \(F\) a field, \(V\) a \(n\)-dimensional \(F\)-vector space, and \(f(x)=x^n+\sum_{i=0}^{n-1}a_ix^i\).
Index: 20
Kind: Theo

Smaller0: \(W\) is \(T\)-invariant
Greater0: \([T]_{\mathcal{B} }\) is block upper triangular \(\left(\begin{array}{ll}A&B\\0&D\end{array}\right)\), where \(A\in\text{Mat}_k(F)\) and \(A=[T|_W]_{\mathcal B_W}\)
Page: 61
Chapter: 4
Section: 1
Variables: \(k<n\in\mathbb N\), \(F\) a field, \(W\subseteq V\), \(k\)- and \(n\)-dimensional \(F\)-vector spaces, and \(\mathcal B_W=\{v_1,\dots,v_k\}\), \(\mathcal B_V=\{v_1,\dots,v_n\}\) basis
Index: 22
Kind: Theo
Smaller to greater: iff
Hide middle: yes

Smaller0: \(m_{(T+W),(v+W)}(x)\)
Greater0: \(m_{T,v}(x)\)
Page: 61
Chapter: 4
Section: 1
Variables: \(F\) a field,
\(V\) a finite dimensional \(F\)-vector space,
\(T\in\text{Hom}(V)\),
\(W\subseteq V\)
\(v\in V\).
Index: 23
Kind: Lem
Smaller to greater: divides
Assuming: \(W\) is \(T\)-invariant
Greater to smaller: is a multiple of

Q: [...] [...] [minimal polynomials]
A: \(m_{(T+W)}(x)\); divides; \(m_{T}(x)\)
Page: 61
Chapter: 4
Section: 1
Variables: \(F\) a field
\(V\) a finite dimensional \(F\)-vector space
\(T\in\text{Hom}(V)\)
\(W\subseteq V\) \(T\)-invariant and \(v\in V\).
Index: 24
Kind: Cor

Q: Equivalently:
-[...]
-[...].
A: \[\left\{\sum_{i=1}^kp_i(T)(v_i)\mid p_i(x)\in F[x]\right\}\]; the smallest \(T\)-invariant subspace of \(V\)-that contains \(\mathcal A\)
Page: 61
Chapter: 4
Section: 1
Variables: \(F\) a field, \(V\) a \(F\)-vector space, \(T\in\text{Hom}(V)\) and \(\mathcal A=\{v_1,\dots,v_k\}\subseteq V\) .
Index: 25, 26
Kind: Def, Ex
Name: The \(T\)-span of \(\mathcal A\)
Extra: que sont p_i, v_i

Question: The dimension of the subspace \(T\)-generated by \(\{w\}\) is
Answer: \(\text{deg}\ m_{T,w}(x)\).
Page: 62
Chapter: 4
Section: 1
Variables: \(F\) a field
\(V\) a \(F\)-vector space
\(T\in\text{Hom}(V)\)
\(w\in V\).
Index: 27
Kind: Lem

Q: [...] [...] [minimal and characteristic polynomials].
A: \(m_T(x)\); divides; \(c_T(x)\)
Page: 62
Chapter: 4
Section: 1
Variables: \(F\) a field
\(V\) a \(F\)-vector space
\(T\in\text{Hom}(V)\)
Index: 28 (a)
Kind: Theo

Q: [...] is [...].
A: Every irreducible factor of \(c_T(x)\); a factor of \(m_T(x)\)
Page: 62
Chapter: 4
Section: 1
Variables: \(F\) a field, \(V\) a \(F\)-vector space, \(T\in\text{Hom}(V)\).
Index: 28 (b)
Kind: Theo

Name: The \(T\)-invariant complement of \(W\)
Definition: \(W'\) such that \(W\oplus W'\) is a \(T\)-invariant direct sum.
Page: 64
Chapter: 4
Section: 2
Variables: \(F\) a field
\(V\) a \(F\)-vector space of dimension \(n\).
\(T\in\text{Hom}_F(V,V)\).
W subseteq V ?
Index: 1
Kind: Def

Smaller0: \(m_T(x)=c_T(x)=g_0(x)g_1(x)\) with \(\gcd (g_0,g_1)>1\)
Greater0: \(g_0(T)(V)\)
Page: 65
Chapter: 4
Section: 2
Variables: \(\mathbb F\) a field, \(V\) a \(\mathbf F\)-vector space and \(T\in\text{Hom}_F(V,V)\). Let \(g_0,g_1\in \mathbb F[x]\)
Index: 3
Kind: Ex
Greater1: \(T\)-invariant subspace without direct sum
Connect Greater0: is

Smaller0: \(m_T(x)=c_T(x)=\Pi_{}g_i(x)\) with \(\gcd (g_i,g_j)=1\) for \(i\ne j\)
Greater0: \(\{\text{ker}(g_i(T))\mid i\}\)
Page: 66
Chapter: 4
Section: 2
Variables: \(\mathbb F\) a field, \(V\) a \(\mathbf F\)-vector space, \(T\in\text{Hom}_F(V,V)\) and \(g_1,\dots,g_k\in \mathbb F[x]\)
Kind: Theo
Hide middle: yes
Greater1: a \(T\)-invariant direct sum decomposition of \(V\)
Connect Greater0: is

Smaller0: \(W\) is \(T\)-invariant
Greater0: \(\bigoplus \left( W\cap \text{ker}(p_i(T)^{e_i})\right)\) with \(m_T(x)\) having irreduccible decomposition \(\Pi p_i(x)^{e_i}\)
Page: 67
Chapter: 4
Section: 2
Variables: \(\mathbb F\) a field, \(V\) a \(\mathbf F\)-vector space, \(T\in\text{Hom}_F(V,V)\) and \(W\) a subspace of \(V\).
Index: 5
Kind: Cor
Prefix Greater: it can be decomposed as the \(T\)-invariant direct sum:

Smaller0: \(m_{T,w_1}(x)=m_T(x)\)
Greater0: there exists \(w_2\in V\) such that \(V\) is the \(T\)-invariant direct sum of the spaces generated by \(w_1\) and \(w_2\)
Page: 68
Chapter: 4
Section: 2
Variables: \(\mathbb F\) a field, \(V\) a \(\mathbf F\)-vector space, \(T\in\text{Hom}_F(V,V)\) and \(w_1\in V\).
Index: 6
Kind: Lem
Hide middle: yes
Smaller1: the \(T\)-invariant space generated by \(w_1\) is proper
Smaller3: there exists \(w_2\in V\) such that \(\{w_1,w_2\}\) \(T\)-generate \(V\)

Smaller0: \(V\) is \(T\)-generated by a single element
Greater0: \(m_T(x)=c_T(x)\)
Page: 63
Chapter: 4
Section: 1
Variables: \(n\in\mathbb N\),
\(F\) a field,
\(V\) a \(F\)-vector space of dimension \(n\)
\(T\in\text{Hom}_F(V,V)\).
Index: 29
Kind: Cor
Smaller to greater: iff

Smaller0: \(m_{T,w_1}(x)=m_T(x)\)
Greater0: the space \(T\)-generated by \(w_1\) has a \(T\)-invariant complement
Page: 70
Chapter: 4
Section: 2
Variables: \(\mathbb F\) a field, \(V\) a \(\mathbf F\)-vector space, \(T\in\text{Hom}_F(V,V)\) and \(w_1\in V\).
Index: 7
Kind: Theo
Hide middle: yes

Smaller0: each \(W_i\) is \(T\)-invariant
Greater0: a \(T\)-invariant direct sum
Page: 63
Chapter: 4
Section: 2
Variables: \(F\) a field,
\(V\) a \(F\)-vector space of dimension \(n\).
\(T\in\text{Hom}_F(V,V)\)
\(W_1,\dots,W_k\) subspaces of \(V\) with \(V=W_1\oplus \dots\oplus W_k\)
Index: 1
Kind: Def
Prefix Greater: \(W_1,\dots,W_k\) is

Smaller0: \(R\) is commutative
Greater0: a right module, defined by \(m\cdot r=rm\)
Page: 30
Chapter: 2
Section: 5
Variables: \(R\) a ring and \(M\) a left \(R\)-module
Index: 1
Kind: Def
Prefix Greater: \(M\) is