Lect24 - Chapter 4. Linear Transformations Math1111 Matrix...

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Unformatted text preview: Chapter 4. Linear Transformations Math1111 Matrix Representations Further Example 1 Example 1 . Let V , W be vector spaces with ordered bases E and F respectively. Let L : V → W be a linear transformation, A be its matrix representation r.t. E and F . Prove the following: (i) w ∈ L ( V ) if and only if [ w ] F ∈ c ( A ) . (ii) v ∈ ker ( L ) if and only if [ v ] E ∈ N ( A ) . (iii) dimker ( L ) = the nullity of A & dim L ( V ) = rank ( A ) . Chapter 4. Linear Transformations Math1111 Matrix Representations Further Example 1 Example 1 . Let V , W be vector spaces with ordered bases E and F respectively. Let L : V → W be a linear transformation, A be its matrix representation r.t. E and F . Prove the following: (i) w ∈ L ( V ) if and only if [ w ] F ∈ c ( A ) . (ii) v ∈ ker ( L ) if and only if [ v ] E ∈ N ( A ) . (iii) dimker ( L ) = the nullity of A & dim L ( V ) = rank ( A ) . Proof of (i): "Only if" part: Let w ∈ L ( V ) . Then L ( v ) = w for some v ∈ V ⇒ [ w ] F = A [ v ] E ⇒ [ w ] F ∈ Span ( a 1 , ··· , a n ) where A = ( a 1 ··· a n ) ⇒ [ w ] F ∈ c ( A ) . Chapter 4. Linear Transformations Math1111 Matrix Representations Further Example 1 Example 1 . Let V , W be vector spaces with ordered bases E and F respectively. Let L : V → W be a linear transformation, A be its matrix representation r.t. E and F . Prove the following: (i) w ∈ L ( V ) if and only if [ w ] F ∈ c ( A ) . (ii) v ∈ ker ( L ) if and only if [ v ] E ∈ N ( A ) . (iii) dimker ( L ) = the nullity of A & dim L ( V ) = rank ( A ) . Proof of (i): "If" part: Let w ∈ W satisfy [ w ] F ∈ c ( A ) . Then [ w ] F = A x for some x = ( x 1 ··· x n ) T ∈ R n . Let E = [ v 1 , ··· , v n ] . Take v = x 1 v 1 + ··· + x n v n , then [ L ( v )] F = [ w ] F (Why?) Hence, L ( v ) = w . Chapter 4. Linear Transformations Math1111 Matrix Representations Further Example 1 Example 1 . Let V , W be vector spaces with ordered bases E and F respectively. Let L : V → W be a linear transformation, A be its matrix representation r.t. E and F . Prove the following: (i) w ∈ L ( V ) if and only if [ w ] F ∈ c ( A ) . (ii) v ∈ ker ( L ) if and only if [ v ] E ∈ N ( A ) . (iii) dimker ( L ) = the nullity of A & dim L ( V ) = rank ( A ) . Theorem Let L : V → W be a linear transformation of vector spaces. Then, dimker ( L ) + dim L ( V ) = dim V . Chapter 4. Linear Transformations Math1111 Matrix Representations Further Example 2 Example . Verify that the dimension formula holds true for the linear transformation L : V → W defined by L ( v ) = . Chapter 4. Linear Transformations Math1111 Matrix Representations Further Example 2 Example . Verify that the dimension formula holds true for the linear transformation L : V → W defined by L ( v ) = ....
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This note was uploaded on 09/06/2010 for the course MATH MATH1111 taught by Professor Forgot during the Fall '08 term at HKU.

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Lect24 - Chapter 4. Linear Transformations Math1111 Matrix...

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