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.

$2.1$ 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.

$2.2$ R(T)= span of T(basis in V)

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

$\mathrm{R}\mathrm{(}\mathrm{T}\mathrm{)}=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}(T(\beta \mathrm{)}\mathrm{)}=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}(T(v_1),T(v_2),\dots ,T(v_n))$.

$2.3$ Dimension theorem

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

$\mathrm{n}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}(T)+\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(T)=\dim (V).$

$2.4$ 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}.

$2.5$ 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).

$2.6$ ∀ $w_1,w_2,\dots ,w_n=$ exactly one T (basis),

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

$2.7$ 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.

$2.8$ 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 $a$.

$2.9$ 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.

$2.10$ 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 $a$.

$2.11$ [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.

$2.12$ 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 $a_1,a_2,\dots ,a_k$ be scalars. Then

$A\left(\sum _{i=1}^k{a}_i{B}_i\right)=\sum _{i=1}^k{a}_{i}A{B}_i$

and

$\left(\sum _{i=1}^k{a}_i{C}_i\right)A=\sum _{i=1}^k{a}_i{C}_{i}A$.

$2.13$ 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) $u_j=A{v}_j$
(b) $v_j=B{e}_j$, where $e_j$ is the jth standard vector of F^p.

$2.14$ [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

$[T(u){]}_{\gamma }=[T{]}_{\beta }^{\gamma }[u{]}_{\beta }$.

$2.15$ 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 $L_{I_n}=I_{F^n}$.

$2.16$ 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.

$2.17$ 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.

$2.18$ [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.

$2.19$ 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.

$2.20$ ??

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 $\mathrm{\Phi }$: $\mathcal{ℒ}$(V,W)→M_m×n(F), defined by $\mathrm{\Phi }(T)=[T{]}_{\beta }^{\gamma }$ 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.

$2.21$ Φ_β is an isomorphism

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

$2.22$ ??

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

$2.23$ [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

$[T{]}_{{\beta }^{\prime }}=Q^{-1}[T{]}_{\beta }Q$.

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.

$2.27$ 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.

$2.28$ {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 ${\mathrm{C}}^{\mathrm{\infty }}$.

$2.29$ derivative of exponential function

For any exponential function $f(t)=e^{ct},f^{\prime }(t)=c{e}^{ct}$.

$2.30$ {e^-at} is a basis of N(p(D+aI))

The solution space for the differential equation,

$y^{\prime }+a_{0}y=0$

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.

$2.31$ $e^{ct}$ 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.

$2.32$ 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^∞.

$2.33$ e^c_it is linearly independent with each other (c_i are distinct)

Given n distinct complex numbers $c_1,c_2,\dots ,c_n$, the set of exponential functions $\{e^{c_{1}t},e^{c_{2}t},\dots ,e^{c_{n}t}\}$ is linearly independent.

Corollary. For any nth-order homogeneous linear differential equation with constant coefficients, if the auxiliary polynomial has n distinct zeros $c_1,c_2,\dots ,c_n$, then $\{e^{c_{1}t},e^{c_{2}t},\dots ,e^{c_{n}t}\}$ 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

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

is a basis for the solution space of the equation.

$2.34$ 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 $n_1,n_2,\dots ,n_k$ are positive integers and $c_1,c_2,\dots ,c_n$ are distinct complex numbers, the following set is a basis for the solution space of the equation:

$\{e^{c_{1}t},t{e}^{c_{1}t},\dots ,t^{n_1-1}{e}^{c_{1}t},\dots ,e{c}_{k}t,t{e}^{c_{k}t},\dots ,t^{n_k-1}{e}^{c_{k}t}\}$.

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

$A=\left(\begin{array}{cc}a& b\\ c& d\\ \end{array}\right)$

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

$\det \left(\begin{array}{c}u+kv\\ w\\ \end{array}\right)=\det \left(\begin{array}{c}u\\ w\\ \end{array}\right)+k\det \left(\begin{array}{c}v\\ w\\ \end{array}\right)$

and

$\det \left(\begin{array}{c}w\\ u+kv\\ \end{array}\right)=\det \left(\begin{array}{c}w\\ u\\ \end{array}\right)+k\det \left(\begin{array}{c}w\\ v\\ \end{array}\right)$

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

$A^{-1}=\frac{1}{\det (A)}\left(\begin{array}{cc}{A}_{22}& -A_{12}\\ -A_{21}& A_{11}\\ \end{array}\right)$

Diagonalization

Characteristic polynomial of a linear operator/matrix

$5.1$ 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, $\beta =v_1,v_2,\dots ,v_n$ 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 $1\le j\le n$.

$5.2$ 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

$5.3$ 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.

$5.4$ 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).

$5.5$ vi to λi⇔vi is linearly independent

Let T be a linear operator on a vector space V, and let ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_k,$ be distinct eigenvalues of T. If $v_1,v_2,\dots ,v_k$ are eigenvectors of t such that ${\lambda }_i$ corresponds to $v_i$ ($1\le i\le k$), then {$v_1,v_2,\dots ,v_k$} is linearly independent.

$5.6$ characteristic polynomial splits

The characteristic polynomial of any diagonalizable linear operator splits.

$5.7$ 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 $1\le \dim (E_{\lambda })\le m$.

$5.8$ S = S₁ ∪ S₂ ∪ ...∪ S_k is linearly independent

Let T be a linear operator on a vector space V, and let ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_k,$ be distinct eigenvalues of T. For each $i=1,2,\dots ,k,$ 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.

$5.9$ ⇔T is diagonalizable

Let T be a linear operator on a finite-dimensional vector space V that the characteristic polynomial of T splits. Let ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_k$ 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 $basi{s}^2$ 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.

$6.1$ 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) $⟨x,y+z⟩=⟨x,y⟩+⟨x,z⟩.$
(b) $⟨x,cy⟩=\overline{c}⟨x,y⟩.$
(c) $⟨x,\mathit{0}⟩=⟨\mathit{0},x⟩=0.$
(d) $⟨x,x⟩=0$ if and only if $x=\mathit{0}.$
(e) If$⟨x,y⟩=⟨x,z⟩$ for all $x\in$ V, then $y=z$.

$6.2$ 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) $\Vert cx\Vert =\left|c\right|\cdot \Vert x\Vert$.
(b) $\Vert x\Vert =0$ if and only if $x=0$. In any case, $\Vert x\Vert \ge 0$.
(c)(Cauchy-Schwarz In equality)$|⟨x,y⟩|\le \Vert x\Vert \cdot \Vert y\Vert$.
(d)(Triangle Inequality)$\Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert$.

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

$6.3$ span of orthogonal subset

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

$y=\sum _{i=1}^n\frac{⟨y,v_i⟩}{\Vert v_i{\Vert }^2}{v}_i$

$6.4$ Gram-Schmidt process

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

$v_k=w_k-\sum _{j=1}^{k-1}\frac{⟨w_k,v_j⟩}{\Vert v_j{\Vert }^2}{v}_j$

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

$6.5$ orthonormal basis

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

$x=\sum _{i=1}^n⟨x,v_i⟩v_i$.

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

$6.6$ 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 $\{v_1,v_2,\dots ,v_k\}$ is an orthornormal basis for W, then

$u=\sum _{i=1}^k⟨y,v_i⟩v_i$.

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, $\Vert y-x\Vert \ge \Vert y-u\Vert$, and this inequality is an equality if and onlly if $x=u$.

$6.7$ properties of orthonormal set

Suppose that $S=\{v_1,v_2,\dots ,v_k\}$ is an orthonormal set in an $n$-dimensional inner product space V. Than
(a) S can be extended to an orthonormal basis $\{v_1,v_2,\dots ,v_k,v_{k+1},\dots ,v_n\}$ for V.
(b) If W=span(S), then $S_1=\{v_{k+1},v_{k+2},\dots ,v_n\}$ 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

$6.8$ 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 $\mathrm{g}\mathrm{(}\mathrm{x}\mathrm{)}=\mathrm{⟨}\mathrm{x},\mathrm{y}\mathrm{⟩}$ for all $x$∈ V.

$6.9$ 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 $⟨\mathrm{T}\mathrm{(}\mathrm{x}\mathrm{)},\mathrm{y}\mathrm{⟩}=\mathrm{⟨}\mathrm{x},{\mathrm{T}}^{*}\mathrm{(}\mathrm{y}\mathrm{)}\mathrm{⟩}$ for all $x,y$ ∈ V. Furthermore, T* is linear

$6.10$ [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

$[T^{*}{]}_{\beta }=[T{]}_{\beta }^{*}$.

$6.11$ 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)*=$\overline{c}$ 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)*=$\overline{c}$ A* for any $c$∈ F;
(c) (AB)*=B*A*;
(d) A**=A;
(e) I*=I.

$6.12$ 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 $\Vert A{x}_0-Y\Vert \le \Vert Ax-y\Vert$ for all x∈ Fⁿ

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

$⟨Ax,y{⟩}_m=⟨x,A*y{⟩}_n$

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.

$6.13$ 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