Theorems and definitions in linear algebra

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 in F and each pair of elements $x,y$ in "V", $a(x+y)=ax+ay$.
*(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", $(a+b)x=ax+bx$."

Vector spaces

ubspaces

Linear combinations

ystems of linear equations

Linear dependence

Linear independence

Bases

Dimension

Linear transformations and matrices

Linear transformations

Null spaces

Ranges

The matrix representation of a linear transformation

Composition of linear transformations

Matrix multiplication

Invertibility

Isomorphisms

=The change-of-coordinates matrix=

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.

$\{color\{Blue\}~2.1\}$ N(T)&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 Vand 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 is a basis for V, then ::.

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

Let V and W be vector spaces, and let T: V→W be linear. If V is finite-dimensional, then ::::::$mbox\{nullity\}(T)+mbox\{rank\}(T)=dim(V).$

=$\{color\{Blue\}~2.4\}$ 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).

=$\{color\{Blue\}~2.6\}$ ∀ $\{w\_1,w\_2...w\_n\}=$ exactly one T(basis),=

Let V and W be vector space over F, and suppose that $\{v\_1,\; v\_2,...,v\_n\}$ is a basis for V. For $w\_1,\; w\_2,...w\_n$ in W, there exists exactly one linear transformation T: V→W such that $mbox\{T\}(v\_i)=w\_i$ for $i=1,2,...n.$ Corollary. Let V and W be vector spaces, and suppose that V has a finite basis $\{v\_1,v\_2,...,v\_n\}$. If U, T: V→W are linear and $U(v\_i)=T(v\_i)$ for $i=1,2,...,n,$ 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", $ambox\{T\}+mbox\{U\}$ 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) and :(b) for all scalars $a$.

$\{color\{Blue\}~2.9\}$ commutative law of linear operator

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 :::::::.

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 $a\_1,a\_2,...,a\_k$ be scalars. Then :::::::$ABigg(sum\_\{i=1\}^k\; a\_iB\_iBigg)=sum\_\{i=1\}^k\; a\_iAB\_i$and:::::::$Bigg(sum\_\{i=1\}^k\; a\_iC\_iBigg)A=sum\_\{i=1\}^k\; a\_iC\_iA$.

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

Let A be an m×n matrix and B be an n×p matrix. For each $j\; (1le\; jle\; p)$ 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 ::::::::.

$\{color\{Blue\}~2.15\}$ 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) . (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, . (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 is invertible. Furthermore,

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 W 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 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 . Then (a) $Q$ is invertible. (b) For any $vin$ V, .

=$\{color\{Blue\}~2.23\}$ [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 :::::::.

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.

$\{color\{Blue\}~2.24\}$

$\{color\{Blue\}~2.25\}$

$\{color\{Blue\}~2.26\}$

=$\{color\{Blue\}~2.27\}$ "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 $mbox\{C\}^infty$.

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

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

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

The solution space for the differential equation, ::::$y\text{'}+a\_0y=0$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 $c\_1,\; c\_2,...,c\_n$, the set of exponential functions $\{e^\{c\_1t\},e^\{c\_2t\},...,e^\{c\_nt\}\}$ is linearly independent.

Corollary. For any nth-order homogeneous linear differential equation with constant coefficients, if the auxiliary polynomial has n distinct zeros $c\_1,\; c\_2,\; ...,\; c\_n$, then $\{e^\{c\_1t\},e^\{c\_2t\},...,e^\{c\_nt\}\}$ 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 :::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 :::::$(t-c\_1)^n\_1(t-c\_2)^n\_2...(t-c\_k)^n\_k,$ where $n\_1,\; n\_2,...,n\_k$ are positive integers and $c\_1,\; c\_2,\; ...,\; c\_n$ are distinct complex numbers, the following set is a basis for the solution space of the equation: :::$\{e^\{c\_1t\},\; te^\{c\_1t\},...,t^\{n\_1-1\}e^\{c\_1t\},...,e\{c\_kt\},te^\{c\_kt\},..,t^\{n\_k-1\}e^\{c\_kt\}\}$.

Elementary matrix operations and systems of linear equations

Elementary matrix operations

Elementary matrix

Rank of a matrix

Matrix inverses

ystem of linear equations

Determinants

"If"::::::::: "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":::::::

"and"

:::::::

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" ::::::::

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, 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 $1le\; j\; le\; n$.

=$\{color\{Blue\}~5.2\}$ eigenvalue⇔det("A"-λ"I"n)=0=

Let "A"∈M_n×n("F"). Then a scalar λ is an eigenvalue of "A" if and only if det("A"-λ"I"_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 alinear operator on a vector space V, and let $lambda\_1,lambda\_2,...,lambda\_k,$ be distinct eigenvalues of T. If $v\_1,v\_2,...,v\_k$ are eigenvectors of t such that $lambda\_i$ corresponds to $v\_i$ ($1le\; ile\; k$), then {$v\_1,v\_2,...,v\_k$} 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 haveing multiplicity $m$. Then $1\; ledim(E\_\{lambda\})le\; m$.

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

Let T e a linear operator on a vector space V, and let $lambda\_1,lambda\_2,...,lambda\_k,$ be distinct eigenvalues of T. For each $i=1,2,...,k,$ let $S\_i$ be a finite linearly indenpendent subset of the eigenspace $E\_\{lambda\_i\}$. Then $S=S\_1cup\; S\_2\; cup...cup\; S\_k$ is a linearly indenpendent 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 $lambda\_1,lambda\_2,...,lambda\_k$ 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 is an ordered basis for $E\_\{lambda\_i\}$ for each $i$, then 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, orthogonal, unit vector, orthonormal, normalizing.

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

Let V be an inner product space. Then for x,y,zin V and c in f, the following staements are true. (a) $langle\; x,y+z\; angle=langle\; x,y\; angle+langle\; x,z\; angle.$ (b) (c) $langle\; x,mathit\{0\}\; angle=langlemathit\{0\},x\; angle=0.$ (d) $langle\; x,x\; angle=0$ if and only if $x=mathit\{0\}.$ (e) If$langle\; x,y\; angle=langle\; x,z\; angle$ for all $xin$ V, then $y=z$.

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

Let V be an inner product space over F. Then for all x,yin V and cin 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 In equality)$|langle\; x,y\; angle|le|x|cdot|y|$. (d)(Triangle Inequality)$|x+y|le|x|+|y|$.

orthonormal basis,Gram-schmidtprocess,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,...,v_k} be an orthogonal subset of V consisting of nonzero vectors. If $y$∈span(S), then ::::::$y=sum\_\{i=1\}^n\{langle\; y,v\_i\; angle\; over\; |v\_i|^2\}v\_i$

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

Let V be an inner product space and S=$\{w\_1,w\_2,...,w\_n\}$ be a linearly independent subset of V. DefineS'=$\{v\_1,v\_2,...,v\_n\}$, where $v\_1=w\_1$ and ::::::$v\_k=w\_k-sum\_\{j=1\}^\{k-1\}\{langle\; w\_k,\; v\_j\; angleover|v\_j|^2\}v\_j$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 =$\{v\_1,v\_2,...,v\_n\}$ and x∈V, then ::::::$x=sum\_\{i=1\}^nlangle\; x,v\_i\; angle\; v\_i$.

Corollary. Let V be a finite-dimensional inner product space with an orthonormal basis β =$\{v\_1,v\_2,...,v\_n\}$. Let T be a linear operator on V, and let A= [T] _β. Then for any $i$ and $j$, $A\_\{ij\}=langle\; T(v\_j),\; v\_i\; angle$.

$\{color\{Blue\}~6.6\}$ 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 $u$∈W^⊥ such that $y=u+z$. Furthermore, if $\{v\_1,v\_2,...,v\_k\}$ is an orthornormal basis for W, then ::::::$u=sum\_\{i=1\}^klangle\; y,v\_i\; angle\; v\_i$.S={v_1,v_2,...,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 $S=\{v\_1,v\_2,...,v\_k\}$ is an orthonormal set in an $n$-dimensional inner product space V. Than (a) S can be extended to an orthonormal basis $\{v\_1,\; v\_2,\; ...,v\_k,v\_\{k+1\},...,v\_n\}$ for V. (b) If W=span(S), then $S\_1=\{v\_\{k+1\},v\_\{k+2\},...,v\_n\}$ 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 $m\{g\}(x)=langle\; x,\; y\; angle$ 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 $langle\; m\{T\}(x),y\; angle=langle\; x,\; m\{T\}^*(y)\; angle$ 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 :::::.

$\{color\{Blue\}~6.11\}$ 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)*= 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")*= "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 :::::$langle\; Ax,\; y\; angle\; \_m\; =langle\; x,\; A*y\; angle\; \_n$

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) Ther vector $s$ is the only solutin to $Ax=b$ that lies in R(L_"A"*); that is , if $u$ satisfies $(AA*)u=b$, then $s=A*u$.

Canonical forms

References

* Linear Algebra 4th edition, by Stephen H. Friedberg Arnold J. Insel and Lawrence E. spence ISBN7040167336
* Linear Algebra 3rd edition, by Serge Lang (UTM) ISBN0387964126