Chapter 2 Linear Mapping and Transformations, Eigenvectors and Eigenvalues
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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
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= ··· ··· ··· ·×n is named the matrix representation of linear mapping A with respect to bases ε1 , ε2 , · · · , εn and η1 , η2 , · · · , ηm .
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.
V3 is another 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
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 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
Theorem 2.1.4 Defined operations as the above, the following statements holds.
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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 )
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
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 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
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
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
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
Chapter 2 Linear Mapping and Transformations, Eigenvectors and Eigenvalues