矩阵论第二章

Chapter 2 Linear Mapping and Transformations, Eigenvectors and Eigenvalues Linear Mapping, Linear Transformation and Matrix Representation
Actually, A’s j −th column is constructed by the coordinate of A(εj ), a11 a12 · · · a1n a21 a22 · · · a2n A= ··· ··· ··· ··· . am1 am2 · · · amn A ∈ P m×n is named the matrix representation of linear mapping A with respect to bases ε1 , ε2 , · · · , εn and η1 , η2 , · · · , ηm .
Chapter 2 Linear Mapping and Transformations, Eigenvectors and Eigenvalues Linear Mapping, Linear Transformation and Matrix Representation
Definition 2.1.1 Let V1 , V2 are two vector spaces on field P , A is a mapping from V1 to V2 . A is a linear mapping (or linear operator ) if it holds, A(α + β ) = A(α) + A(β ), A(k α) = k A(α), ∀α, β ∈ V1 , ∀α ∈ V1 , k ∈ P .
Chapter 2 Linear Mapping and Transformations, Eigenvectors and Eigenvalues
Matrix Theory
Wang Liping ( w²)
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P. R. China, email: wlpmath@yahoo.com.cn A ‘‡d L TEX^‡?
Chapter 2 Linear Mapping and Transformations, Eigenvectors and Eigenvalues Linear Mapping, Linear Transformation and Matrix Representation
Let V1 , V2 be two vector spaces on field P . Arbitrarily given A, B ∈ L(V1 , V2 ), k ∈ P and define (A + B)(α) = A(α) + B(α), (k A)(α) = k A(α). ∀α ∈ V1
Theorem 2.1.4 Defined operations as the above, the following statements holds.
1 2 3
If A, B ∈ L(V1 , V2 ), then A + B ∈ L(V1 , V2 ). For any k ∈ P and A ∈ L(V1 , V2 ), then k B ∈ L(V1 , V2 ). If A ∈ L(V1 , V2 ) and C ∈ L(V2 , V3 ), then CA ∈ L(V1 , V3 )
Theorem 2.1.5 L(V1 , V2 ) is a vector space with above defined operations.
Chapter 2 Linear Mapping and Transformations, Eigenvectors and Eigenvalues Linear Mapping, Linear Transformation and Matrix Representation
Theorem 2.1.1 If A ∈ L(V1 , V2 ), then the following statements hold.
1 2 3
A(0) = 0. A(−α) = −A(α). If α1 , α2 , · · · , αm are linearly dependent in V1 , then A(α1 ), A(α2 ), · · · , A(αm ) are also linearly dependent in V2 . If A is one-to-one (or bijective) linear mapping, then α1 , α2 , · · · , αm ∈ V1 and A(α1 ), A(α2 ), · · · , A(αm ) ∈ V2 have the same linear dependence.
Chapter 2 Linear Mapping and Transformations, Eigenvectors and Eigenvalues Linear Mapping, Linear Transformation and Matrix Representation
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Chapter 2 Linear Mapping and Transformations, Eigenvectors and Eigenvalues Linear Mapping, Linear Transformation and Matrix Representation Change of Representing Matrix and Similarity Unitary (Orthogonal) Transformations Isomorphism Eigenvalue and Eigenvector Diagonalization
The set of all linear mapping from V1 to V2 is denoted by L(V1 , V2 ).
Chapter 2 Linear Mapping and Transformations, Eigenvectors and Eigenvalues Linear Mapping, Linear Transformation and Matrix Representation
V3 is anLeabharlann Baiduther vector space, if C ∈ L(V2 , V3 ), then CA is defined (CA)(α) = C(A(α)), ∀α ∈ V1 .
Chapter 2 Linear Mapping and Transformations, Eigenvectors and Eigenvalues Linear Mapping, Linear Transformation and Matrix Representation
Chapter 2 Linear Mapping and Transformations, Eigenvectors and Eigenvalues Linear Mapping, Linear Transformation and Matrix Representation
Suppose that ε1 , ε2 , · · · , εn and η1 , η2 , · · · , ηm are bases of V1 and V2 respectively. A ∈ L(V1 , V2 ) and A ∈ P m×n is its matrix representation with respect to the above bases. Given α ∈ V1 , let
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Chapter 2 Linear Mapping and Transformations, Eigenvectors and Eigenvalues Linear Mapping, Linear Transformation and Matrix Representation
Theorem 2.1.2 Let A, B ∈ L(V1 , V2 ), and ε1 , ε2 , · · · , εn is a basis of V1 . If A(εi ) = B(εi ), i = 1, 2, · · · , n, then A = B. Remark 2.1.1 A linearly mapping is uniquely determined by the images on basis vectors. Theorem 2.1.3 Suppose that ε1 , ε2 , · · · , εn is a basis of V1 , and α1 , α2 , · · · , αn are arbitrary given vectors in V2 , there exists a unique A ∈ L(V1 , V2 ) such that A(εi ) = αi , i = 1, 2, · · · , n .
Suppose that V1 , V2 are two vector spaces on field P with n and m dimensions, ε1 , ε2 , · · · , εn and η1 , η2 , · · · , ηm are bases of V1 and V2 respectively. A ∈ L(V1 , V2 ) holds A(ε1 ) = a11 η1 + a21 η2 + · · · + am1 ηm A(ε2 ) = a12 η1 + a22 η2 + · · · + am2 ηm ··· ··· ··· A(εn ) = a1n η1 + a2n η2 + · · · + amn ηm It can be simply denoted by A(ε1 , ε2 , · · · , εn ) = (A(ε1 ), A(ε2 ), · · · , A(εn )) = (η1 , η2 , · · · , ηm )A.
Chapter 2 Linear Mapping and Transformations, Eigenvectors and Eigenvalues
1
Chapter 2 Linear Mapping and Transformations, Eigenvectors and Eigenvalues Linear Mapping, Linear Transformation and Matrix Representation Change of Representing Matrix and Similarity Unitary (Orthogonal) Transformations Isomorphism Eigenvalue and Eigenvector Diagonalization
n m
α=
i =1
xi εi ,
A(α) =
i =1
yi ηi ,
then y = Ax , where x = (x1 , x2 , · · · , xn )T and (y1 , y2 , · · · , ym )T are coordinates of α and A(α).
Chapter 2 Linear Mapping and Transformations, Eigenvectors and Eigenvalues Linear Mapping, Linear Transformation and Matrix Representation