Entropy power inequality

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

Vector spaces

A vector space( or linear space) V over a number field² F consists of a set on which two operations (called addition and scalar multiplication, respectively) are defined so, that for each pair of elements x, y, in V there is a unique element x + y in V, and for each element a in F and each element x in V there is a unique element ax in V, such that the following conditions hold.

• (VS 1) For all $x,y$ in V, $x+y=y+x$ (commutativity of addition).
• (VS 2) For all $x,y,z$ in V, $(x+y)+z=x+(y+z)$ (associativity of addition).
• (VS 3) There exists an element in V denoted by $0$ such that $x+0=x$ for each $x$ in V.
• (VS 4) For each element $x$ in V there exists an element $y$ in V such that $x+y=0$.
• (VS 5) For each element $x$ in V, $1x=x$.
• (VS 6) For each pair of element $a,b$ in F and each element $x$ in V, $(ab)x=a(bx)$.
• (VS 7) For each element $a$ in F and each pair of elements $x,y$ in V, $a(x+y)=ax+ay$.
• (VS 8) For each pair of elements $a,b$ in F and each pair of elements $x$ in V, $(a+b)x=ax+bx$.

Subspaces

A subspace W of a vector space V over a field F is a subset of V which also has the properties that W is closed under scaler addition an multiplication. That is, For all x, y in W, x and y are in V and for any c in F, $cx+y$ is in W.

Linear combinations

Linear combination

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 polynomialTemplate:Disambiguation needed, to the power of a complex number, exponential function.

${\color {Blue}~2.1}$ 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 V and W be vector spaces, and let T: V→W be linear. Then T is one-to-one if and only if N(T)={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.

(c) rank(T) = dim(V).

${\color {Blue}~2.6}$ ∀ ${w_{1},w_{2},\ldots ,w_{n}}=$ 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 ve 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 $a$.

(c) I_mA=AI_m.

(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

$A{\Bigg (}\sum _{i=1}^{k}a_{i}B_{i}{\Bigg )}=\sum _{i=1}^{k}a_{i}AB_{i}$

and

${\Bigg (}\sum _{i=1}^{k}a_{i}C_{i}{\Bigg )}A=\sum _{i=1}^{k}a_{i}C_{i}A$.

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

Let A be an m×n matrix and B be an n×p matrix. For each $j(1\leq j\leq p)$ 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, $\mathrm {C} =[L_{A}]_{\beta }^{\gamma }$.
(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.

${\color {Blue}~2.27}$ 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,

$y'+a_{0}y=0$

is of dimension 1 and has $\{e^{-a_{0}t}\}$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 $\{e^{c_{1}t},e^{c_{2}t},\ldots ,e^{c_{n}t}\}$ 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 $\{e^{c_{1}t},e^{c_{2}t},\ldots ,e^{c_{n}t}\}$ 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

$\beta =\{e^{c_{1}t},e^{c_{2}t},\ldots ,e^{c_{n}t}\}$

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

$(t-c_{1})_{1}^{n}(t-c_{2})_{2}^{n}\cdots (t-c_{k})_{k}^{n},$

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:

$\{e^{c_{1}t},te^{c_{1}t},\ldots ,t^{n_{1}-1}e^{c_{1}t},\ldots ,e{c_{k}t},te^{c_{k}t},\ldots ,t^{n_{k}-1}e^{c_{k}t}\}$.

Elementary matrix operations and systems of linear equations

Rank of a matrix

The rank of a matrix A is the dimension of the column space of A.

Determinants

If

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

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 $1\leq j\leq n$.

${\color {Blue}~5.2}$ 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 $v_{i}$ ($1\leq i\leq k$), 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 $1\leq \dim(E_{\lambda })\leq m$.

${\color {Blue}~5.8}$ 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, adjointTemplate:Dn, Frobenius inner product, complex/real inner product space, norm, length, conjugate linear, orthogonal, perpendicular, orthogonal, unit vector, orthonormal, normalization.

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

Let V be an inner product space. Then for x,y,z\in V and c \in f, the following staements 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\in V and c\in F, the following statements are true.
(a) $\|cx\|=|c|\cdot \|x\|$.
(b) $\|x\|=0$ if and only if $x=0$. In any case, $\|x\|\geq 0$.
(c)(Cauchy-Schwarz In equality)$|\langle x,y\rangle |\leq \|x\|\cdot \|y\|$.
(d)(Triangle Inequality)$\|x+y\|\leq \|x\|+\|y\|$.

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, $\|y-x\|\geq \|y-u\|$, 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 $\|Ax_{0}-Y\|\leq \|Ax-y\|$ 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