Theorems and definitions in linear algebra

From Wikipedia, the free encyclopedia

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

1 Vector spaces
2 Linear transformations and matrices
3 Elementary matrix operations and systems of linear equations
4 Determinants
5 Diagonalization
6 Inner Product Spaces
7 Canonical forms
8 References

[edit] 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, $1 x = x$ .
(VS 6) For each pair of element a in F and each pair of elements $x, y$ in V, $a (x + y) = a x + a y$ .
(VS 7) For each element $a$ in F and each pair of elements $x, y$ in V, $a (x + y) = a x + a y$ .
(VS 8) For each pair of elements $a, b$ in F and each pair of elements $x$ in V, $(a + b) x = a x + b x$ .

[edit] Vector spaces

[edit] Subspaces

[edit] Linear combinations

[edit] Systems of linear equations

[edit] Linear dependence

[edit] Linear independence

[edit] Bases

[edit] Dimension

[edit] 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.

[edit] ${\color{Blue}~2.1}$ 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.

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

Let V and W be vector spaces, and let T: V→W be linear. If $β = v 1, v 2,..., v n$ is a basis for V, then

R(T) = span(T (β)) = span(T (v 1), T (v 2),..., T (v n))

[edit] ${\color{Blue}~2.3}$ 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).

[edit] ${\color{Blue}~2.4}$ 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}.

[edit] ${\color{Blue}~2.5}$ 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.

[edit] ${\color{Blue}~2.6}$ ∀ $w 1, w 2 ... w n =$ 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 $T(v i) = w i$ 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.

[edit] ${\color{Blue}~2.7}$ 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 T + 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.

[edit] ${\color{Blue}~2.8}$ 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

[edit] ${\color{Blue}~2.9}$ 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.

[edit] ${\color{Blue}~2.10}$ 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 $a$ .

[edit] ${\color{Blue}~2.11}$ [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.

[edit] ${\color{Blue}~2.12}$ 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 $a 1, a 2,..., a k$ be scalars. Then

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

and

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

[edit] ${\color{Blue}~2.13}$ 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) $u j = A v j$
(b) $v j = B e j$ , where $e j$ is the jth standard vector of F^p.

[edit] ${\color{Blue}~2.14}$ [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

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

[edit] ${\color{Blue}~2.15}$ 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, $\mbox{C}=[L_A]_\beta^\gamma$ .
(e) If W is an n×p matrix, then L_AE=L_AL_E.
(f ) If m=n, then $L_{I_n}=I_{F^n}$ .

[edit] ${\color{Blue}~2.16}$ 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.

[edit] ${\color{Blue}~2.17}$ 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.

[edit] ${\color{Blue}~2.18}$ [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.

[edit] ${\color{Blue}~2.19}$ 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.

[edit] ${\color{Blue}~2.20}$ ??

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 $~\Phi(T)=[T]_\beta^\gamma$ 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.

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

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

[edit] ${\color{Blue}~2.22}$ ??

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, $~[v]_\beta=Q[v]_{\beta'}$ .

[edit] ${\color{Blue}~2.23}$ [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.

[edit] ${\color{Blue}~2.24}$

[edit] ${\color{Blue}~2.25}$

[edit] ${\color{Blue}~2.26}$

[edit] ${\color{Blue}~2.27}$ 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.

[edit] ${\color{Blue}~2.28}$ {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 $\mbox{C}^\infty$ .

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

For any exponential function $f (t) = e c t, f'(t) = c e c t$ .

[edit] ${\color{Blue}~2.30}$ {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 $\{e^{-a_0t}\}$ as a basis.

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

[edit] ${\color{Blue}~2.31}$ $e c t$ 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.

[edit] ${\color{Blue}~2.32}$ 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^∞.

[edit] ${\color{Blue}~2.33}$ 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

$\beta=\{e^{c_1t},e^{c_2t},...,e^{c_nt}\}$

is a basis for the solution space of the equation.

[edit] ${\color{Blue}~2.34}$ 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:

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

[edit] Elementary matrix operations and systems of linear equations

[edit] Elementary matrix operations

[edit] Elementary matrix

[edit] Rank of a matrix

[edit] Matrix inverses

[edit] System of linear equations

[edit] 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 $a d - b c$ .

＊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

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

[edit] Diagonalization

Characteristic polynomial of a linear operator/matrix

[edit] ${\color{Blue}~5.1}$ 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, $β = v 1, v 2,..., v n$ is an ordered basis of eigenvectors of T, and D = [T]_β then D is a diagonal matrix and $D j j$ is the eigenvalue corresponding to $v j$ for $1\le j \le n$ .

[edit] ${\color{Blue}~5.2}$ 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

[edit] ${\color{Blue}~5.3}$ 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.

[edit] ${\color{Blue}~5.4}$ 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).

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

Let T be alinear operator on a vector space V, and let $λ 1,λ 2,...,λ k,$ be distinct eigenvalues of T. If $v 1, v 2,..., v k$ are eigenvectors of t such that $λ i$ corresponds to $v i$ ( $1\le i\le k$ ), then { $v 1, v 2,..., v k$ } is linearly independent.

[edit] ${\color{Blue}~5.6}$ characteristic polynomial splits

The characteristic polynomial of any diagonalizable linear operator splits.

[edit] ${\color{Blue}~5.7}$ 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 $1 \le\dim(E_{\lambda})\le m$ .

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

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

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

Let T be a linear operator on a finite-dimensional vector space V that the characteristic polynomial of T splits. Let $λ 1,λ 2,...,λ k$ be the distinct eigenvalues of T. Then
(a) T is diagonalizable if and only if the multiplicity of $λ i$ is equal to $\dim(E_{\lambda_i})$ for all $i$ .
(b) If T is diagonalizable and $β i$ 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 $b a s i s 2$ for V consisting of eigenvectors of T.

Test for diagonlization

[edit] 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.

[edit] ${\color{Blue}~6.1}$ 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 Failed to parse (Cannot write to or create math output directory): x=\mathit{0}.
(e) If $\langle x,y\rangle=\langle x,z\rangle$ for all $x\in$ V, then $y = z$ .

[edit] ${\color{Blue}~6.2}$ 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, $\|x\|\ge0$ .
(c)(Cauchy-Schwarz In equality) $|\langle x,y\rangle|\le\|x\|\cdot\|y\|$ .
(d)(Triangle Inequality) $\|x+y\|\le\|x\|+\|y\|$ .

orthonormal basis,Gram-schmidtprocess,Fourier coefficients,orthogonal complement,orthogonal projection

[edit] ${\color{Blue}~6.3}$ 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

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

[edit] ${\color{Blue}~6.4}$ 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

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

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

[edit] ${\color{Blue}~6.5}$ 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

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

Corollary. Let V be a finite-dimensional inner product space with an orthonormal basis β = ${v 1, v 2,..., v n}$ . Let T be a linear operator on V, and let A=[T]_β. Then for any $i$ and $j$ , $A_{ij}=\langle T(v_j), v_i\rangle$ .

[edit] ${\color{Blue}~6.6}$ 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

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

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

[edit] ${\color{Blue}~6.7}$ 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

[edit] ${\color{Blue}~6.8}$ 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.

[edit] ${\color{Blue}~6.9}$ 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

[edit] ${\color{Blue}~6.10}$ [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

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

[edit] ${\color{Blue}~6.11}$ 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.

[edit] ${\color{Blue}~6.12}$ 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.

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

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

[edit] Canonical forms

[edit] 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

Categories: Linear algebra

Theorems and definitions in linear algebra

From Wikipedia, the free encyclopedia

Contents

[edit] Vector spaces

[edit] Vector spaces

[edit] Subspaces

[edit] Linear combinations

[edit] Systems of linear equations

[edit] Linear dependence

[edit] Linear independence

[edit] Bases

[edit] Dimension

[edit] Linear transformations and matrices

[edit] N(T)&R(T) are subspaces

[edit] R(T)= span of T(basis in V)

[edit] Dimension Theorem

[edit] one-to-one ⇔ N(T)={0}

[edit] one-to-one ⇔ onto ⇔ rank(T)=dim(V)

[edit] ∀ w1,w2...wn = exactly one T(basis),

[edit] T is vector space

[edit] linearity of matrix representation of linear transformation

[edit] commutative law of linear operator

[edit] law of linear operator

[edit] [UT]αγ=[U]βγ[T]αβ

[edit] law of matrix

[edit] law of column multiplication

[edit] [T(u)]γ=[T]βγ[u]β

[edit] laws of LA

[edit] A(BC)=(AB)C

[edit] T-1is linear

[edit] [T-1]γβ=([T]βγ)-1

[edit] V is isomorphic to W ⇔ dim(V)=dim(W)

[edit] ??

[edit] Φβ is an isomorphism

[edit] ??

[edit] [T]β'=Q-1[T]βQ

[edit]

[edit]

[edit]

[edit] p(D)(x)=0 (p(D)∈C∞)⇒ x(k)exists (k∈N)

[edit] {solutions}= N(p(D))

[edit] derivative of exponential function

[edit] {e-at} is a basis of N(p(D+aI))

[edit] ect is a solution

[edit] dim(N(p(D)))=n

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

[edit] general solution of homogeneous linear differential equation

[edit] Elementary matrix operations and systems of linear equations

[edit] Elementary matrix operations

[edit] Elementary matrix

[edit] Rank of a matrix

[edit] Matrix inverses

[edit] System of linear equations

[edit] Determinants

[edit] Diagonalization

[edit] diagonalizable⇔basis of eigenvector

[edit] eigenvalue⇔det(A-λIn)=0

[edit] characteristic polynomial

[edit] υ to λ⇔υ∈N(T-λI)

[edit] vi to λi⇔vi is linearly independent

[edit] characteristic polynomial splits

[edit] 1≤dim(Eλ)≤m

[edit] S=S1∪S2∪...∪Sk is linearly indenpendent

[edit] ⇔T is diagonalizable

[edit] Inner Product Spaces

[edit] properties of linear product

[edit] law of norm

[edit] span of orthogonal subset

[edit] Gram-Schmidt process

[edit] orthonormal basis

[edit] W⊥ by orthonormal basis

[edit] properties of orthonormal set

[edit] linear functional representation inner product

[edit] definition of T*

[edit] [T*]β=[T]*β

[edit] properties of T*

[edit] Least squares approximation

[edit] Minimal solutions to systems of linear equations

[edit] Canonical forms

[edit] References

[edit] ${\color{Blue}~2.1}$ N(T)&R(T) are subspaces

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

[edit] ${\color{Blue}~2.3}$ Dimension Theorem

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

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

[edit] ${\color{Blue}~2.6}$ ∀ $w 1, w 2 ... w n =$ exactly one T(basis),

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

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

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

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

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

[edit] ${\color{Blue}~2.12}$ law of matrix

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

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

[edit] ${\color{Blue}~2.15}$ laws of L_A

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

[edit] ${\color{Blue}~2.17}$ T^-1is linear

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

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

[edit] ${\color{Blue}~2.20}$ ??

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

[edit] ${\color{Blue}~2.22}$ ??

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

[edit] ${\color{Blue}~2.24}$

[edit] ${\color{Blue}~2.25}$

[edit] ${\color{Blue}~2.26}$

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

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

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

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

[edit] ${\color{Blue}~2.31}$ $e c t$ is a solution

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

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

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

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

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

[edit] ${\color{Blue}~5.3}$ characteristic polynomial

[edit] ${\color{Blue}~5.4}$ υ to λ⇔υ∈N(T-λI)

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

[edit] ${\color{Blue}~5.6}$ characteristic polynomial splits

[edit] ${\color{Blue}~5.7}$ 1≤dim(Eλ)≤m

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

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

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

[edit] ${\color{Blue}~6.2}$ law of norm

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

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

[edit] ${\color{Blue}~6.5}$ orthonormal basis

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

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

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

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

[edit] ${\color{Blue}~6.10}$ [T]_β=[T]_β

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

[edit] ${\color{Blue}~6.12}$ Least squares approximation

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