Theorems and definitions in linear algebra

This article collects the main theorems and definitions in linear algebra.

Vector Spaces

Let $V$ be a set on which two operations (vector addition and scalar multiplication) are defined. If the listed axioms are satisfied for every $\vec u$ , $\vec v$ , and $\vec w$ in $V$ and every scalar $c$ and $d$ , then $V$ is called a vector space:

Addition:

$\vec u + \vec v\text{ is in }V\text{.}$
$\vec u + \vec v = \vec v + \vec u$
$\vec u + (\vec v + \vec w) = (\vec u + \vec v) + \vec w$
$V\text{ has a }\mathbf{zero}\text{ }\mathbf{vector}\text{ }\vec 0\text{ such that for every }\vec u\text{ in }V\text{, }\vec u + \vec 0 = \vec u$
$\text{For every }\vec u\text{ in }V\text{, there is a vector in }V\text{ denoted by }-\vec u\text{ such that }\vec u + (-\vec u) = \vec 0\text{.}$

Scalar Multiplication:

$c\vec u\text{ is in }V\text{.}$
$c(\vec u + \vec v) = c\vec u + c\vec v$
$(c + d)\vec u = c\vec u + d\vec u$
$c(d\vec u) = (cd)\vec u$
$1(\vec u) = \vec u$

Subspaces

If $W$ is a nonempty subset of a vector space $V$ , then $W$ is a subspace of $V$ if and only if the following closure conditions hold:

$\text{If }\vec u\text{ and }\vec v\text{ are in }W\text{, then }\vec u + \vec v\text{ is in }W\text{.}$
$\text{If }\vec u\text{ is in }W\text{ and }c\text{ is any scalar, then }c\vec u\text{ is in }W\text{.}$

Linear combinations

A vector $\vec v$ in a vector space $V$ is called a linear combination of the vectors $\vec u_1, \vec u_2, \cdots , \vec u_k$ in $V$ if $\vec v$ can be written in the form $\vec v = c_1\vec u_1 + c_2 \vec u_2 + \cdots + c_k\vec u_k$ , where $c_1, c_2, \cdots , c_k$ are scalars.

Systems of linear equations

A system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.

Cramer's Rule

If a system of $n$ linear equations in $n$ variables has a coefficient matrix with a nonzero determinant | $A$ |, then the solution of the system is given by

x_1 = \frac{\det(A_1)}{\det(A)}, \qquad x_2 = \frac{\det(A_2)}{\det(A)} , \qquad \dots , \qquad x_n = \frac{\det(A_n)}{\det(A)}

where $A i$ is the matrix $A$ with the $i$ -th column of $A$ replaced by the column of constants in the system of equations.

Linear dependence

A set of vectors $\{\vec{v_1}, \vec{v_2}, \cdots, \vec{v_k}\}$ in a vector space $V$ is linearly dependent if there exists a set of scalars, ${x 1, x 2, ..., x k$ }, not all zero, so that the vector equation $x_1\vec{v_1}+x_2\vec{v_2}+\cdots+x_k\vec{v_k} = \vec{0}$ has a solution.

Linear independence

A set of vectors $\{\vec{v_1}, \vec{v_2}, \cdots, \vec{v_k}\}$ in a vector space $V$ is linearly independent if the vector equation $x_1\vec{v_1}+x_2\vec{v_2}+\cdots+x_k\vec{v_k} = \vec{0}$ has only the trivial solution, $x_1 = x_2 = \cdots = x_k = 0$ .

Bases

A set of vectors $S = \{\vec v_1, \vec v_2, \cdots , \vec v_n\}$ in a vector space $V$ is called a basis if the following conditions are true:

$S$ spans $V$ .
$S$ is linearly independent.

Linear transformations and matrices

Change of coordinate matrix
Clique
Coordinate vector relative to a basis
Dimension theorem
Dominance relation
Identity matrix
Identity transformation
Incidence matrix
Inverse of a linear transformation
Inverse of a matrix
Invertible linear transformation
Isomorphic vector spaces
Isomorphism
Kronecker delta
Left-multiplication transformation
Linear operator
Linear transformation
Matrix representing a linear transformation
Nullity of a linear transformation
Null space
Ordered basis
Product of matrices
Projection on a subspace
Projection on the x-axis
Range
Rank of a linear transformation
Reflection about the x-axis
Rotation
Similar matrices
Standard ordered basis for $F_n$
Standard representation of a vector space with respect to a basis
Zero transformation

P.S. coefficient of the differential equation, differentiability of complex function,vector space of functionsdifferential operator, auxiliary polynomial, to the power of a complex number, exponential function.

Definition of a Linear Transformation

Let $V$ and $W$ be vector spaces. The function $T:V\to W$ is called a linear transformation of $V$ into $W$ if the following two properties are true for all $\vec u$ and $\vec v$ in $V$ and for any scalar $c$ .

$T(\vec u + \vec v) = T(\vec u) + T(\vec v)$
$T(c\vec u) = cT(\vec u)$

${\color{Blue}~2.1}$ N(T) and R(T) are subspaces

Let V and W be vector spaces and I: V → W be linear. Then N(T) and R(T) are subspaces of V and W, respectively.

${\color{Blue}~2.2}$ R(T)= span of T(basis in V)

Let V and W be vector spaces, and let T: V→W be linear. If $\beta={v_1,v_2,\ldots,v_n}$ is a basis for V, then

\mathrm{R(T)}=\mathrm{span}(T(\beta\mathrm{))}=\mathrm{span}({T(v_1),T(v_2),\ldots,T(v_n)})

${\color{Blue}~2.3}$ Dimension theorem

Let V and W be vector spaces, and let T: V → W be linear. If V is finite-dimensional, then

\mathrm{nullity}(T)+\mathrm{rank}(T)=\dim(V).

${\color{Blue}~2.4}$ one-to-one ⇔ N(T) = {0}

Let $T:V\to W$ be a linear transformation. Then $T$ is one-to-one if and only if $\operatorname{ker}(T) = \{\vec 0\}$ .

${\color{Blue}~2.5}$ one-to-one ⇔ onto ⇔ rank(T) = dim(V)

Let V and W be vector spaces of equal (finite) dimension, and let T:V → W be linear. Then the following are equivalent.

(a) T is one-to-one.

(b) T is onto.

${\color{Blue}~2.6}$ ∀ ${w_1,w_2,\ldots,w_n}=$ exactly one T (basis),

Let V and W be vector space over F, and suppose that ${v_1, v_2,\ldots,v_n}$ is a basis for V. For $w_1, w_2,\ldots,w_n$ in W, there exists exactly one linear transformation T: V→W such that $\mathrm{T}(v_i)=w_i$ for $i=1,2,\ldots,n.$
Corollary. Let V and W be vector spaces, and suppose that V has a finite basis ${v_1,v_2,\ldots,v_n}$ . If U, T: V→W are linear and $U(v_i)=T(v_i)$ for $i=1,2,\ldots,n,$ then U=T.

${\color{Blue}~2.7}$ T is vector space

Let V and W be vector spaces over a field F, and let T, U: V→W be linear.

(a) For all

a

∈ F,

a\mathrm{T}+\mathrm{U}

is linear.

(b) Using the operations of addition and scalar multiplication in the preceding definition, the collection of all linear transformations form V to W is a vector space over F.

${\color{Blue}~2.8}$ linearity of matrix representation of linear transformation

Let V and W be finite-dimensional vector spaces with ordered bases β and γ, respectively, and let T, U: V→W be linear transformations. Then

(a)

[T+U]_\beta^\gamma=[T]_\beta^\gamma+[U]_\beta^\gamma

and

(b)

[aT]_\beta^\gamma=a[T]_\beta^\gamma

for all scalars

a

${\color{Blue}~2.9}$ composition law of linear operators

Let V,W, and Z be vector spaces over the same field f, and let T:V→W and U:W→Z be linear. then UT:V→Z is linear.

${\color{Blue}~2.10}$ law of linear operator

Let v be a vector space. Let T, U₁, U₂ ∈ $\mathcal{L}$ (V). Then
(a) T(U₁+U₂)=TU₁+TU₂ and (U₁+U₂)T=U₁T+U₂T
(b) T(U₁U₂)=(TU₁)U₂
(c) TI=IT=T
(d) $a$ (U₁U₂)=( $a$ U₁)U₂=U₁( $a$ U₂) for all scalars $a$ .

${\color{Blue}~2.11}$ [UT]_α^γ=[U]_β^γ[T]_α^β

Let V, W and Z be finite-dimensional vector spaces with ordered bases α β γ, respectively. Let T: V→W and U: W→Z be linear transformations. Then

[UT]_\alpha^\gamma=[U]_\beta^\gamma[T]_\alpha^\beta

Corollary. Let V be a finite-dimensional vector space with an ordered basis β. Let T,U∈ $\mathcal{L}$ (V). Then [UT]_β=[U]_β[T]_β.

${\color{Blue}~2.12}$ law of matrix

Let A be an m×n matrix, B and C be n×p matrices, and D and E be q×m matrices. Then

(a) A(B+C)=AB+AC and (D+E)A=DA+EA.

(b)

a

(AB)=(

a

A)B=A(

a

B) for any scalar

a

(d) If V is an n-dimensional vector space with an ordered basis β, then [I_v]_β=I_n.

Corollary. Let A be an m×n matrix, B₁,B₂,...,B_k be n×p matrices, C₁,C₁,...,C₁ be q×m matrices, and $a_1,a_2,\ldots,a_k$ be scalars. Then

A\Bigg(\sum_{i=1}^k a_iB_i\Bigg)=\sum_{i=1}^k a_iAB_i

and

\Bigg(\sum_{i=1}^k a_iC_i\Bigg)A=\sum_{i=1}^k a_iC_iA

${\color{Blue}~2.13}$ law of column multiplication

Let A be an m×n matrix and B be an n×p matrix. For each $j (1\le j\le p)$ let $u_j$ and $v_j$ denote the jth columns of AB and B, respectively. Then
(a) $u_j=Av_j$
(b) $v_j=Be_j$ , where $e_j$ is the jth standard vector of F^p.

${\color{Blue}~2.14}$ [T(u)]_γ=[T]_β^γ[u]_β

Let V and W be finite-dimensional vector spaces having ordered bases β and γ, respectively, and let T: V→W be linear. Then, for each u ∈ V, we have

[T(u)]_\gamma=[T]_\beta^\gamma[u]_\beta

${\color{Blue}~2.15}$ laws of L_A

Let A be an m×n matrix with entries from F. Then the left-multiplication transformation L_A: Fⁿ→F^m is linear. Furthermore, if B is any other m×n matrix (with entries from F) and β and γ are the standard ordered bases for Fⁿ and F^m, respectively, then we have the following properties.
(a) $[L_A]_\beta^\gamma=A$ .
(b) L_A=L_B if and only if A=B.
(c) L_A+B=L_A+L_B and L_{$a$ A}= $a$ L_A for all $a$ ∈F.
(d) If T:Fⁿ→F^m is linear, then there exists a unique m×n matrix C such that T=L_C. In fact, $\mathrm{C}=[L_T]_\beta^\gamma$ .
(e) If W is an n×p matrix, then L_AE=L_AL_E.
(f ) If m=n, then $L_{I_n}=I_{F^n}$ .

${\color{Blue}~2.16}$ A(BC)=(AB)C

Let A,B, and C be matrices such that A(BC) is defined. Then A(BC)=(AB)C; that is, matrix multiplication is associative.

${\color{Blue}~2.17}$ T⁻¹is linear

Let V and W be vector spaces, and let T:V→W be linear and invertible. Then T⁻¹: W →V is linear.

${\color{Blue}~2.18}$ [T⁻¹]_γ^β=([T]_β^γ)⁻¹

Let V and W be finite-dimensional vector spaces with ordered bases β and γ, respectively. Let T:V→W be linear. Then T is invertible if and only if $[T]_\beta^\gamma$ is invertible. Furthermore, $[T^{-1}]_\gamma^\beta=([T]_\beta^\gamma)^{-1}$

Lemma. Let T be an invertible linear transformation from V to W. Then V is finite-dimensional if and only if W is finite-dimensional. In this case, dim(V)=dim(W).

Corollary 1. Let V be a finite-dimensional vector space with an ordered basis β, and let T:V→V be linear. Then T is invertible if and only if [T]_β is invertible. Furthermore, [T⁻¹]_β=([T]_β)⁻¹.

Corollary 2. Let A be an n×n matrix. Then A is invertible if and only if L_A is invertible. Furthermore, (L_A)⁻¹=L_A⁻¹.

${\color{Blue}~2.19}$ V is isomorphic to W ⇔ dim(V)=dim(W)

Let W and W be finite-dimensional vector spaces (over the same field). Then V is isomorphic to W if and only if dim(V)=dim(W).

Corollary. Let V be a vector space over F. Then V is isomorphic to Fⁿ if and only if dim(V)=n.

${\color{Blue}~2.20}$ ??

Let V and W be finite-dimensional vector spaces over F of dimensions n and m, respectively, and let β and γ be ordered bases for V and W, respectively. Then the function $~\Phi$ : $\mathcal{L}$ (V,W)→M_m×n(F), defined by $~\Phi(T)=[T]_\beta^\gamma$ for T∈ $\mathcal{L}$ (V,W), is an isomorphism.

Corollary. Let V and W be finite-dimensional vector spaces of dimension n and m, respectively. Then $\mathcal{L}$ (V,W) is finite-dimensional of dimension mn.

${\color{Blue}~2.21}$ Φ_β is an isomorphism

For any finite-dimensional vector space V with ordered basis β, Φ_β is an isomorphism.

${\color{Blue}~2.22}$ ??

Let β and β' be two ordered bases for a finite-dimensional vector space V, and let $Q=[I_V]_{\beta'}^\beta$ . Then
(a) $Q$ is invertible.
(b) For any $v\in$ V, $~[v]_\beta=Q[v]_{\beta'}$ .

${\color{Blue}~2.23}$ [T]_β'=Q⁻¹[T]_βQ

Let T be a linear operator on a finite-dimensional vector space V,and let β and β' be two ordered bases for V. Suppose that Q is the change of coordinate matrix that changes β'-coordinates into β-coordinates. Then

~[T]_{\beta'}=Q^{-1}[T]_\beta Q

Corollary. Let A∈M_n×n(F), and le t γ be an ordered basis for Fⁿ. Then [L_A]_γ=Q⁻¹AQ, where Q is the n×n matrix whose jth column is the jth vector of γ.

Principal Axes Theorem

For a conic whose equation is $ax^2 + bxy + cy^2 + dx + ey + f = 0$ , the rotation given by $X=PX'$ eliminates the $xy$ -term if $P$ is an orthogonal matrix, with $\left\vert P\right\vert = 1$ , that diagonalizes $A$ . That is,

P^TAP = \begin{bmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{bmatrix}

┌────────────────────────────────────────────────────────────────────────────────────────────────────┘where $\lambda_1$ and $\lambda_2$ are eigenvalues of $A$ . The equation of the rotated conic is given by

\lambda_1(x')^2 + \lambda_2(y')^2 + \begin{bmatrix} d & e\\ \end{bmatrix}PX' + f = 0

${\color{Blue}~2.27}$ p(D)(x)=0 (p(D)∈C^∞)⇒ x^(k)exists (k∈N)

Any solution to a homogeneous linear differential equation with constant coefficients has derivatives of all orders; that is, if $x$ is a solution to such an equation, then $x^{(k)}$ exists for every positive integer k.

${\color{Blue}~2.28}$ {solutions}= N(p(D))

The set of all solutions to a homogeneous linear differential equation with constant coefficients coincides with the null space of p(D), where p(t) is the auxiliary polynomial with the equation.

Corollary. The set of all solutions to s homogeneous linear differential equation with constant coefficients is a subspace of $\mathrm{C}^\infty$ .

${\color{Blue}~2.29}$ derivative of exponential function

For any exponential function $f(t)=e^{ct}, f'(t)=ce^{ct}$ .

${\color{Blue}~2.30}$ {e^−at} is a basis of N(p(D+aI))

The solution space for the differential equation,

y'+a_0y=0

is of dimension 1 and has $\{e^{-a_0t}\}$ as a basis.

Corollary. For any complex number c, the null space of the differential operator D-cI has { $e^{ct}$ } as a basis.

${\color{Blue}~2.31}$ $e^{ct}$ is a solution

Let p(t) be the auxiliary polynomial for a homogeneous linear differential equation with constant coefficients. For any complex number c, if c is a zero of p(t), then to the differential equation.

${\color{Blue}~2.32}$ dim(N(p(D)))=n

For any differential operator p(D) of order n, the null space of p(D) is an n_dimensional subspace of C^∞.

Lemma 1. The differential operator D-cI: C^∞ to C^∞ is onto for any complex number c.

Lemma 2 Let V be a vector space, and suppose that T and U are linear operators on V such that U is onto and the null spaces of T and U are finite-dimensional, Then the null space of TU is finite-dimensional, and

dim(N(TU))=dim(N(U))+dim(N(U)).

Corollary. The solution space of any nth-order homogeneous linear differential equation with constant coefficients is an n-dimensional subspace of C^∞.

${\color{Blue}~2.33}$ e^c_it is linearly independent with each other (c_i are distinct)

Given n distinct complex numbers $c_1, c_2,\ldots,c_n$ , the set of exponential functions $\{e^{c_1t},e^{c_2t},\ldots,e^{c_nt}\}$ is linearly independent.

Corollary. For any nth-order homogeneous linear differential equation with constant coefficients, if the auxiliary polynomial has n distinct zeros $c_1, c_2, \ldots, c_n$ , then $\{e^{c_1t},e^{c_2t},\ldots,e^{c_nt}\}$ is a basis for the solution space of the differential equation.

Lemma. For a given complex number c and positive integer n, suppose that (t-c)^n is athe auxiliary polynomial of a homogeneous linear differential equation with constant coefficients. Then the set

\beta=\{e^{c_1t},e^{c_2t},\ldots,e^{c_nt}\}

is a basis for the solution space of the equation.

${\color{Blue}~2.34}$ general solution of homogeneous linear differential equation

Given a homogeneous linear differential equation with constant coefficients and auxiliary polynomial

(t-c_1)^{n_1}(t-c_2)^{n_2}\cdots(t-c_k)^{n_k},

where $n_1, n_2,\ldots,n_k$ are positive integers and $c_1, c_2, \ldots, c_n$ are distinct complex numbers, the following set is a basis for the solution space of the equation:

\{e^{c_1t}, te^{c_1t},\ldots,t^{n_1-1}e^{c_1t},\ldots,e^{c_kt},te^{c_kt},\ldots,t^{n_k-1}e^{c_kt}\}

Definition of an Orthogonal Matrix

A square matrix $P$ is called orthogonal if it is invertible and if

P^{-1} = P^T

Real Spectral Theorem

If $A$ is an $n\times n$ symmetric matrix, then the following properties are true:

$A$ is diagonalizable.
All eigenvalues of $A$ are real.
If $\lambda$ is an eigenvalue of $A$ with multiplicity $k$ , then $\lambda$ has $k$ linearly independent eigenvectors. That is, the eigenspace of $\lambda$ has dimension $k$ .

Also, the set of eigenvalues of $A$ is called the spectrum of $A$ .

Elementary matrix operations and systems of linear equations

Elementary matrix operations

The three elementary row operations are the following:

Interchange two rows.
Multiply a row by a nonzero constant.
Add a multiple of a row to another row.

Elementary matrix

An $n \times n$ matrix is called an elementary matrix if it can be obtained from the identity matrix $I_n$ by a single elementary row operation.

Rank of a matrix

The rank of a matrix A is the number of pivot columns after the reduced row echelon form of A.

Invertible Matrices

$\text{If }A\text{ is }n\text{ × }n\text{, then the following statements are equivalent:}$

$A \text{ is invertible.}$
$A\vec x = \vec b\text{ has a unique solution for every }n \times 1\text{ column matrix }\vec b\text{.}$
$A\vec x = \vec 0\text{ has only the trivial solution.}$
$A\text{ is row-equivalent to }I_{n}\text{.}$
$A\text{ can be written as the product of elementary matrices.}$
$\det(A) \ne 0$
$\operatorname{rk}(A) = n\text{ number of columns.}$
$\operatorname{nul}(A) = 0$
$\text{All of the }n\text{-row vectors of }A\text{ are linearly independent.}$
$\text{All of the }n\text{-column vectors of }A\text{ are linearly independent.}$

Determinants

A = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}

is a 2×2 matrix with entries form a field F, then we define the determinant of A, denoted det(A) or |A|, to be the scalar $ad-bc$ .

＊Theorem 1: linear function for a single row.
＊Theorem 2: nonzero determinant ⇔ invertible matrix

Theorem 1: The function det: M_2×2(F) → F is a linear function of each row of a 2×2 matrix when the other row is held fixed. That is, if $u,v,$ and $w$ are in F² and $k$ is a scalar, then

\det\begin{pmatrix} u + kv\\ w\\ \end{pmatrix} =\det\begin{pmatrix} u\\ w\\ \end{pmatrix} + k\det\begin{pmatrix} v\\ w\\ \end{pmatrix}

and

\det\begin{pmatrix} w\\ u + kv\\ \end{pmatrix} =\det\begin{pmatrix} w\\ u\\ \end{pmatrix} + k\det\begin{pmatrix} w\\ v\\ \end{pmatrix}

Theorem 2: Let A $\in$ M_2×2(F). Then thee deter minant of A is nonzero if and only if A is invertible. Moreover, if A is invertible, then

A^{-1}=\frac{1}{\det(A)}\begin{pmatrix} A_{22}&-A_{12}\\ -A_{21}&A_{11}\\ \end{pmatrix}

Diagonalization

Characteristic polynomial of a linear operator/matrix

${\color{Blue}~5.1}$ diagonalizable⇔basis of eigenvector

A linear operator T on a finite-dimensional vector space V is diagonalizable if and only if there exists an ordered basis β for V consisting of eigenvectors of T. Furthermore, if T is diagonalizable, $\beta= {v_1,v_2,\ldots,v_n}$ is an ordered basis of eigenvectors of T, and D = [T]_β then D is a diagonal matrix and $D_{jj}$ is the eigenvalue corresponding to $v_j$ for $1\le j \le n$ .

${\color{Blue}~5.2}$ eigenvalue⇔det(A-λIn)=0

Let A∈M_n×n(F). Then a scalar λ is an eigenvalue of A if and only if det(A-λI_n)=0

${\color{Blue}~5.3}$ characteristic polynomial

Let A∈Mn×n(F).
(a) The characteristic polynomial of A is a polynomial of degree n with leading coefficient (-1)ⁿ.
(b) A has at most n distinct eigenvalues.

${\color{Blue}~5.4}$ υ to λ⇔υ∈N(T-λI)

Let T be a linear operator on a vector space V, and let λ be an eigenvalue of T.
A vector υ∈V is an eigenvector of T corresponding to λ if and only if υ≠0 and υ∈N(T-λI).

${\color{Blue}~5.5}$ vi to λi⇔vi is linearly independent

Let T be a linear operator on a vector space V, and let $\lambda_1,\lambda_2,\ldots,\lambda_k,$ be distinct eigenvalues of T. If $v_1,v_2,\ldots,v_k$ are eigenvectors of t such that $\lambda_i$ corresponds to $v_i$ ( $1\le i\le k$ ), then { $v_1,v_2,\ldots,v_k$ } is linearly independent.

${\color{Blue}~5.6}$ characteristic polynomial splits

The characteristic polynomial of any diagonalizable linear operator splits.

${\color{Blue}~5.7}$ 1 ≤ dim(Eλ) ≤ m

Let T be alinear operator on a finite-dimensional vectorspace V, and let λ be an eigenvalue of T having multiplicity $m$ . Then $1 \le\dim(E_{\lambda})\le m$ .

${\color{Blue}~5.8}$ S = S₁ ∪ S₂ ∪ ...∪ S_k is linearly independent

Let T be a linear operator on a vector space V, and let $\lambda_1,\lambda_2,\ldots,\lambda_k,$ be distinct eigenvalues of T. For each $i=1,2,\ldots,k,$ let $S_i$ be a finite linearly independent subset of the eigenspace $E_{\lambda_i}$ . Then $S=S_1\cup S_2 \cup\cdots\cup S_k$ is a linearly independent subset of V.

${\color{Blue}~5.9}$ ⇔T is diagonalizable

Let T be a linear operator on a finite-dimensional vector space V that the characteristic polynomial of T splits. Let $\lambda_1,\lambda_2,\ldots,\lambda_k$ be the distinct eigenvalues of T. Then
(a) T is diagonalizable if and only if the multiplicity of $\lambda_i$ is equal to $\dim(E_{\lambda_i})$ for all $i$ .
(b) If T is diagonalizable and $\beta_i$ is an ordered basis for $E_{\lambda_i}$ for each $i$ , then $\beta=\beta_1\cup \beta_2\cup \cup\beta_k$ is an ordered $basis^2$ for V consisting of eigenvectors of T.

Test for diagonlization

Inner product spaces

Inner product, standard inner product on Fⁿ, conjugate transpose, adjoint, Frobenius inner product, complex/real inner product space, norm, length, conjugate linear, orthogonal, perpendicular, unit vector, orthonormal, normalization.

${\color{Blue}~6.1}$ properties of linear product

Let V be an inner product space. Then for x,y,z ∈ V and c ∈ F, the following statements are true.
(a) $\langle x,y+z\rangle=\langle x,y\rangle+\langle x,z\rangle.$
(b) $\langle x,cy\rangle=\bar{c}\langle x,y\rangle.$
(c) $\langle x,\mathit{0}\rangle=\langle\mathit{0},x\rangle=0.$
(d) $\langle x,x\rangle=0$ if and only if $x=\mathit{0}.$
(e) If $\langle x,y\rangle=\langle x,z\rangle$ for all $x\in$ V, then $y=z$ .

${\color{Blue}~6.2}$ law of norm

Let V be an inner product space over F. Then for all x,y ∈ V and c ∈ F, the following statements are true.
(a) $\|cx\|=|c|\cdot\|x\|$ .
(b) $\|x\|=0$ if and only if $x=0$ . In any case, $\|x\|\ge0$ .
(c)(Cauchy-Schwarz Inequality) $|\langle x,y\rangle|\le\|x\|\cdot\|y\|$ .
(d)(Triangle Inequality) $\|x+y\|\le\|x\|+\|y\|$ .

orthonormal basis, Gram–Schmidt process, Fourier coefficients, orthogonal complement, orthogonal projection

${\color{Blue}~6.3}$ span of orthogonal subset

Let V be an inner product space and $S=\{v_1,v_2,\ldots,v_k\}$ be an orthogonal subset of V consisting of nonzero vectors. If $y$ ∈span(S), then

y=\sum_{i=1}^n{\langle y,v_i \rangle \over \|v_i\|^2}v_i

${\color{Blue}~6.4}$ Gram-Schmidt process

Let V be an inner product space and S= $\{w_1,w_2,\ldots,w_n\}$ be a linearly independent subset of V. DefineS'= $\{v_1,v_2,\ldots,v_n\}$ , where $v_1=w_1$ and

v_k=w_k-\sum_{j=1}^{k-1}{\langle w_k, v_j\rangle\over\|v_j\|^2}v_j

Then S' is an orhtogonal set of nonzero vectors such that span(S')=span(S).

${\color{Blue}~6.5}$ orthonormal basis

Let V be a nonzero finite-dimensional inner product space. Then V has an orthonormal basis β. Furthermore, if β = $\{v_1,v_2,\ldots,v_n\}$ and x∈V, then

x=\sum_{i=1}^n\langle x,v_i\rangle v_i

Corollary. Let V be a finite-dimensional inner product space with an orthonormal basis β = $\{v_1,v_2,\ldots,v_n\}$ . Let T be a linear operator on V, and let A=[T]_β. Then for any $i$ and $j$ , $A_{ij}=\langle T(v_j), v_i\rangle$ .

${\color{Blue}~6.6}$ W^⊥ by orthonormal basis

Let W be a finite-dimensional subspace of an inner product space V, and let $y$ ∈V. Then there exist unique vectors $u$ ∈W and $z$ ∈W^⊥ such that $y=u+z$ . Furthermore, if $\{v_1,v_2,\ldots,v_k\}$ is an orthornormal basis for W, then

u=\sum_{i=1}^k\langle y,v_i\rangle v_i

S=\{v_1,v_2,\ldots,v_k\} Corollary. In the notation of Theorem 6.6, the vector $u$ is the unique vector in W that is "closest" to $y$ ; thet is, for any $x$ ∈W, $\|y-x\|\ge\|y-u\|$ , and this inequality is an equality if and onlly if $x=u$ .

${\color{Blue}~6.7}$ properties of orthonormal set

Suppose that $S=\{v_1,v_2,\ldots,v_k\}$ is an orthonormal set in an $n$ -dimensional inner product space V. Than
(a) S can be extended to an orthonormal basis $\{v_1, v_2, \ldots,v_k,v_{k+1},\ldots,v_n\}$ for V.
(b) If W=span(S), then $S_1=\{v_{k+1},v_{k+2},\ldots,v_n\}$ is an orhtonormal basis for W^⊥(using the preceding notation).
(c) If W is any subspace of V, then dim(V)=dim(W)+dim(W^⊥).

Least squares approximation, Minimal solutions to systems of linear equations

${\color{Blue}~6.8}$ linear functional representation inner product

Let V be a finite-dimensional inner product space over F, and let $g$ :V→F be a linear transformation. Then there exists a unique vector $y$ ∈ V such that $\rm{g}(x)=\langle x, y\rangle$ for all $x$ ∈ V.

${\color{Blue}~6.9}$ definition of T*

Let V be a finite-dimensional inner product space, and let T be a linear operator on V. Then there exists a unique function T*:V→V such that $\langle\rm{T}(x),y\rangle=\langle x, \rm{T}^*(y)\rangle$ for all $x,y$ ∈ V. Furthermore, T* is linear

${\color{Blue}~6.10}$ [T]_β=[T]_β

Let V be a finite-dimensional inner product space, and let β be an orthonormal basis for V. If T is a linear operator on V, then

[T^*]_\beta=[T]^*_\beta

${\color{Blue}~6.11}$ properties of T*

Let V be an inner product space, and let T and U be linear operators onV. Then
(a) (T+U)*=T*+U*;
(b) ( $c$ T)*= $\bar c$ T* for any c∈ F;
(c) (TU)*=U*T*;
(d) T**=T;
(e) I*=I.

Corollary. Let A and B be n×nmatrices. Then
(a) (A+B)*=A*+B*;
(b) ( $c$ A)*= $\bar c$ A* for any $c$ ∈ F;
(c) (AB)*=B*A*;
(d) A**=A;
(e) I*=I.

${\color{Blue}~6.12}$ Least squares approximation

Let A ∈ M_m×n(F) and $y$ ∈F^m. Then there exists $x_0$ ∈ Fⁿ such that $(A*A)x_0=A*y$ and $\|Ax_0-Y\|\le\|Ax-y\|$ for all x∈ Fⁿ

Lemma 1. let A ∈ M_m×n(F), $x$ ∈Fⁿ, and $y$ ∈F^m. Then

\langle Ax, y\rangle _m =\langle x, A*y\rangle _n

Lemma 2. Let A ∈ M_m×n(F). Then rank(A*A)=rank(A).

Corollary.(of lemma 2) If A is an m×n matrix such that rank(A)=n, then A*A is invertible.

${\color{Blue}~6.13}$ Minimal solutions to systems of linear equations

Let A ∈ M_m×n(F) and b∈ F^m. Suppose that $Ax=b$ is consistent. Then the following statements are true.
(a) There existes exactly one minimal solution $s$ of $Ax=b$ , and $s$ ∈R(L_A*).
(b) The vector $s$ is the only solution to $Ax=b$ that lies in R(L_A*); that is, if $u$ satisfies $(AA*)u=b$ , then $s=A*u$ .

References

Linear Algebra 4th edition, by Stephen H. Friedberg Arnold J. Insel and Lawrence E. spence ISBN 7-04-016733-6
Linear Algebra 3rd edition, by Serge Lang (UTM) ISBN 0-387-96412-6
Linear Algebra and Its Applications 4th edition, by Gilbert Strang ISBN 0-03-010567-6

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Theorems and definitions in linear algebra

Vector Spaces

Subspaces

Linear combinations

Systems of linear equations

Cramer's Rule

Linear dependence

Linear independence

Bases

Linear transformations and matrices

Definition of a Linear Transformation

N(T) and R(T) are subspaces

R(T)= span of T(basis in V)

Dimension theorem

one-to-one ⇔ N(T) = {0}

one-to-one ⇔ onto ⇔ rank(T) = dim(V)

∀ exactly one T (basis),

T is vector space

linearity of matrix representation of linear transformation

composition law of linear operators

law of linear operator

[UT]αγ=[U]βγ[T]αβ

law of matrix

law of column multiplication

[T(u)]γ=[T]βγ[u]β

laws of LA

A(BC)=(AB)C

T−1is linear

[T−1]γβ=([T]βγ)−1

V is isomorphic to W ⇔ dim(V)=dim(W)

??

Φβ is an isomorphism

??

[T]β'=Q−1[T]βQ

Principal Axes Theorem

p(D)(x)=0 (p(D)∈C∞)⇒ x(k)exists (k∈N)

{solutions}= N(p(D))

derivative of exponential function

{e−at} is a basis of N(p(D+aI))

is a solution

dim(N(p(D)))=n

ecit is linearly independent with each other (ci are distinct)

general solution of homogeneous linear differential equation