Lect25a

# Lect25a - Chapter 4 Linear Transformations Math1111 Matrix...

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Unformatted text preview: Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Bases - Example (Cont’d) Example . Let L : V → W be the linear transformation. Show that there are bases E and F for V and W respectively such that the matrix representation of L r.t. E and F is of the form A =      1 . . . 1      . Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Bases - Example (Cont’d) Example . Let L : V → W be the linear transformation. Show that there are bases E and F for V and W respectively such that the matrix representation of L r.t. E and F is of the form A =      1 . . . 1      . Question How many 1 ’s are there? Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Bases - Example (Cont’d) Example . Let L : V → W be the linear transformation. Show that there are bases E and F for V and W respectively such that the matrix representation of L r.t. E and F is of the form A =      1 . . . 1      . Question How many 1 ’s are there? If such bases exist, then [ L ( x )] F = A [ x ] E . Recall y ∈ L ( V ) if and only if [ y ] F ∈ c ( A ) . What is c ( A ) ? What is its dimension? Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Bases - Example (Cont’d) Example . Let L : V → W be the linear transformation. Show that there are bases E and F for V and W respectively such that the matrix representation of L r.t. E and F is of the form A =      1 . . . 1      . Key : Let M be an m × n matrix. We can find suitable elementary matrices E 1 , ··· , E k , F 1 , ··· , F l so that E 1 ··· E k M F 1 ··· F l =      1 . . . 1      Call Q- 1 = E 1 ··· E k & P = F 1 ··· F l . Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Bases - Example (Cont’d) Example . Let L : V → W be the linear transformation. Show that there are bases E and F for V and W respectively such that the matrix representation of L r.t. E and F is of the form A =      1 ....
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## This note was uploaded on 04/30/2010 for the course MATH 1111 taught by Professor Dr,li during the Spring '10 term at HKU.

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Lect25a - Chapter 4 Linear Transformations Math1111 Matrix...

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