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

Subspaces

Linear combinations

Systems 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 β = v₁, v₂, ..., v_n is a basis for V, then
R(T) = span(T(β)) = span(T(v₁),T(v₂),...,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
nullity(T) + 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).

${\color{Blue}~2.6}$ ∀ w₁, w₂...w_n= exactly one T(basis),

Let V and W be vector space over F, and suppose that v₁, v₂, ..., v_n is a basis for V. For w₁, w₂, ...w_n in W, there exists exactly one linear transformation T: V→W such that 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 v₁, v₂, ..., 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, aT + 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]_β^γ = [T]_β^γ + [U]_β^γ and
:(b)[aT]_β^γ = a[T]_β^γ 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 ∈ 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)=(aU₁)U₂=U₁(aU₂) 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]_α^γ = [U]_β^γ[T]_α^β.

Corollary. Let V be a finite-dimensional vector space with an ordered basis β. Let T,U∈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)=(aA)B=A(aB) 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 a₁, a₂, ..., 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≤j≤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)]_γ = [T]_β^γ[u]_β.

${\color{Blue}~2.15}$ 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]_β^γ = A.
(b) L_A=L_B if and only if A=B.
(c) L_A+B=L_A+L_B and L_aA=aL_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, C = [L_A]_β^γ.
(e) If W is an n×p matrix, then L_AE=L_AL_E.
(f ) If m=n, then L_In = I_Fⁿ.

===${\color{Blue}~2.16}$ 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]_β^γ is invertible. Furthermore, [T⁻¹]_γ^β = ([T]_β^γ)⁻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 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  Φ: ℒ(V,W)→M_m×n(F), defined by  Φ(T) = [T]_β^γ for T∈ℒ(V,W), is an isomorphism.
Corollary. Let V and W be finite-dimensional vector spaces of dimension n and m, respectively. Then 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]_β′^β. Then
(a) Q is invertible.
(b) For any v∈ V,  [v]_β = Q[v]_β′.

===${\color{Blue}~2.23}$ [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]_β′ = Q⁻¹[T]_βQ.

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 C^∞.

${\color{Blue}~2.29}$ 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′ + a₀y = 0 is of dimension 1 and has {e^−a₀t}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 c₁, c₂, ..., c_n, the set of exponential functions {e^c₁t, e^c₂t, ..., 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 c₁, c₂, ..., c_n, then {e^c₁t, e^c₂t, ..., 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
β = {e^c₁t, e^c₂t, ..., 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−c₁)₁ⁿ(t−c₂)₂ⁿ...(t−c_k)_kⁿ,
where n₁, n₂, ..., n_k are positive integers and c₁, c₂, ..., c_n are distinct complex numbers, the following set is a basis for the solution space of the equation:
{e^c₁t, te^c₁t, ..., t^{n₁ − 1}e^c₁t, ..., ec_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

System of linear equations

Determinants

If

$A\; =\; \backslash begin\{pmatrix\}$

a & b \\ c & d \\ \end{pmatrix} is a 2×2matrix with entries form a field F, then we define the determinant of A, denoteddet(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 functiondet: M_2×2(F)→ F is a linear function of each row of a2×2matrix when the other row is held fixed. That is, if u, v, and w are inF²and 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 ∈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 eigenvectorsof T. Furthermore, if T is diagonalizable, β = v₁, v₂, ..., 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 ≤ j ≤ 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 nwith 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 λ₁, λ₂, ..., λ_k, be distinct eigenvalues of T. If v₁, v₂, ..., v_k are eigenvectors of t such that λ_i corresponds to v_i (1 ≤ i ≤ k), then {v₁, v₂, ..., 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 ≤ dim (E_λ) ≤ 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 λ₁, λ₂, ..., λ_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_λi. Then S = S₁ ∪ S₂ ∪ ... ∪ 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 λ₁, λ₂, ..., λ_k be the distinct eigenvalues of T. Then
(a) T is diagonalizable if and only if the multiplicity of λ_i is equal to dim (E_λi) for all i.
(b) If T is diagonalizable and β_i is an ordered basis for E_λi for each i, then β = β₁ ∪ β₂ ∪  ∪ β_k is an ordered basis² 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,z\in V and c \in f, the following staements are true.
(a) ⟨x, y + z⟩ = ⟨x, y⟩ + ⟨x, z⟩.
(b) ⟨x, cy⟩ = c̄⟨x, y⟩.
(c) ⟨x, 0⟩ = ⟨0, x⟩ = 0.
(d) ⟨x, x⟩ = 0 if and only if x = 0.
(e) If⟨x, y⟩ = ⟨x, z⟩ for all x∈ V, then y = z.

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

Let V be an innner product space over F. Then for all x,y\in V and c\in F, the following statements are true.
(a) ∥cx∥ = |c| ⋅ ∥x∥.
(b) ∥x∥ = 0 if and only if x = 0. In any case, ∥x∥ ≥ 0.
(c)(Cauchy-Schwarz In equality)|⟨x,y⟩| ≤ ∥x∥ ⋅ ∥y∥.
(d)(Triangle Inequality)∥x + y∥ ≤ ∥x∥ + ∥y∥.
orthonormal basis,Gram-schmidtprocess,Fourier coefficients,orthogonal complement,orthogonal projection

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

Let V be an innner 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
$$y=\sum_{i=1}^n{\langle y,v_i \rangle \over \|v_i\|^2}v_i$$

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

Let V be an inner product space and S={w₁, w₂, ..., w_n} be a linearly independent subset of V. DefineS'={v₁, v₂, ..., v_n}, where v₁ = w₁ 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 ={v₁, v₂, ..., 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 ={v₁, v₂, ..., 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⟩.

${\color{Blue}~6.6}$ 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 {v₁, v₂, ..., 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,...,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∥ ≥ ∥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 = {v₁, v₂, ..., v_k} is an orthonormal set in an n-dimensional inner product space V. Than
(a) S can be extended to an orthonormal basis {v₁, v₂, ..., v_k, v_k + 1, ..., v_n} for V.
(b) If W=span(S), then S₁ = {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 $\rm{g}(x)=\langle x, y\rangle$ 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^*]_β = [T]_β^*.

${\color{Blue}~6.11}$ 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) (cT)*=c̄ 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) (cA)*=c̄ 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 x₀ ∈ Fⁿ such that (A*A)x₀ = A * y and ∥Ax₀ − Y∥ ≤ ∥Ax − y∥ 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.

${\color{Blue}~6.13}$ 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.

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