Lect22 - Chapter 4. Linear Transformations Math1111 Matrix...

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Unformatted text preview: Chapter 4. Linear Transformations Math1111 Matrix Representations of Linear Transformations Motivation Let L : R 2 → R 3 , L (( a b ) T ) = ( a 0 0 ) T be a linear transformation. Suppose { e 1 , e 2 } is the standard basis of R 2 . Chapter 4. Linear Transformations Math1111 Matrix Representations of Linear Transformations Motivation Let L : R 2 → R 3 , L (( a b ) T ) = ( a 0 0 ) T be a linear transformation. Suppose { e 1 , e 2 } is the standard basis of R 2 . Then, L ( e 1 ) =     1     , L ( e 2 ) =         . Chapter 4. Linear Transformations Math1111 Matrix Representations of Linear Transformations Motivation Let L : R 2 → R 3 , L (( a b ) T ) = ( a 0 0 ) T be a linear transformation. Suppose { e 1 , e 2 } is the standard basis of R 2 . Then, L ( e 1 ) =     1     , L ( e 2 ) =         . ∵ each x ∈ R 2 is of the form x = x 1 e 1 + x 2 e 2 . ∴ L ( x ) = x 1 L ( e 1 )+ x 2 L ( e 2 ) Chapter 4. Linear Transformations Math1111 Matrix Representations of Linear Transformations Motivation Let L : R 2 → R 3 , L (( a b ) T ) = ( a 0 0 ) T be a linear transformation. Suppose { e 1 , e 2 } is the standard basis of R 2 . Then, L ( e 1 ) =     1     , L ( e 2 ) =         . ∵ each x ∈ R 2 is of the form x = x 1 e 1 + x 2 e 2 . ∴ L ( x ) = x 1 L ( e 1 )+ x 2 L ( e 2 ) =     1     x 1 x 2 ! = A x 1 x 2 ! where A =     1     . Chapter 4. Linear Transformations Math1111 Matrix Representations of Linear Transformations Motivation Let L : R 2 → R 3 , L (( a b ) T ) = ( a 0 0 ) T be a linear transformation. Suppose { e 1 , e 2 } is the standard basis of R 2 . Then, L ( e 1 ) =     1     , L ( e 2 ) =         . ∵ each x ∈ R 2 is of the form x = x 1 e 1 + x 2 e 2 . ∴ L ( x ) = x 1 L ( e 1 )+ x 2 L ( e 2 ) =     1     x 1 x 2 ! = A x 1 x 2 ! where A =     1     . What is done? Express a linear transformation in terms of a matrix Chapter 4. Linear Transformations Math1111 Matrix Representations Theorem 4.2.1 Theorem 4.2.1 Let L : R n → R m be a linear transformation. For any x = ( x 1 ··· x n ) T ∈ R n , there is a unique m × n matrix A such that L ( x ) = A x and the j th column of A , a j = L ( e j ) . We call A the standard matrix representation . Chapter 4. Linear Transformations Math1111 Matrix Representations Theorem 4.2.1 Theorem 4.2.1 Let L : R n → R m be a linear transformation. For any x = ( x 1 ··· x n ) T ∈ R n , there is a unique m × n matrix A such that L ( x ) = A x and the j th column of A , a j = L ( e j ) ....
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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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Lect22 - Chapter 4. Linear Transformations Math1111 Matrix...

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