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Lect22 - Chapter 4 Linear Transformations Math1111 Matrix...

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Chapter 4. Linear Transformations Math1111 Matrix Representations of Linear Transformations Motivation Let L : 2 3 , L (( a b ) T ) = ( a 0 0 ) T be a linear transformation. Suppose { e 1 , e 2 } is the standard basis of 2 .
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Chapter 4. Linear Transformations Math1111 Matrix Representations of Linear Transformations Motivation Let L : 2 3 , L (( a b ) T ) = ( a 0 0 ) T be a linear transformation. Suppose { e 1 , e 2 } is the standard basis of 2 . Then, L ( e 1 ) = 1 0 0 , L ( e 2 ) = 0 0 0 .
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Chapter 4. Linear Transformations Math1111 Matrix Representations of Linear Transformations Motivation Let L : 2 3 , L (( a b ) T ) = ( a 0 0 ) T be a linear transformation. Suppose { e 1 , e 2 } is the standard basis of 2 . Then, L ( e 1 ) = 1 0 0 , L ( e 2 ) = 0 0 0 . each x 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 )
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Chapter 4. Linear Transformations Math1111 Matrix Representations of Linear Transformations Motivation Let L : 2 3 , L (( a b ) T ) = ( a 0 0 ) T be a linear transformation. Suppose { e 1 , e 2 } is the standard basis of 2 . Then, L ( e 1 ) = 1 0 0 , L ( e 2 ) = 0 0 0 . each x 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 0 0 0 0 0 x 1 x 2 = A x 1 x 2 where A = 1 0 0 0 0 0 .
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Chapter 4. Linear Transformations Math1111 Matrix Representations of Linear Transformations Motivation Let L : 2 3 , L (( a b ) T ) = ( a 0 0 ) T be a linear transformation. Suppose { e 1 , e 2 } is the standard basis of 2 . Then, L ( e 1 ) = 1 0 0 , L ( e 2 ) = 0 0 0 . each x 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 0 0 0 0 0 x 1 x 2 = A x 1 x 2 where A = 1 0 0 0 0 0 . What is done? Express a linear transformation in terms of a matrix
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Chapter 4. Linear Transformations Math1111 Matrix Representations Theorem 4.2.1 Theorem 4.2.1 Let L : n m be a linear transformation. For any x = ( x 1 ··· x n ) T 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 .
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Chapter 4. Linear Transformations Math1111 Matrix Representations Theorem 4.2.1 Theorem 4.2.1 Let L : n m be a linear transformation. For any x = ( x 1 ··· x n ) T 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 . Proof. Set a j = L ( e j ) where j = 1, ··· , n and A = ( a 1 ··· a n ) ,
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Chapter 4. Linear Transformations Math1111 Matrix Representations Theorem 4.2.1 Theorem 4.2.1 Let L : n m be a linear transformation. For any x = ( x 1 ··· x n ) T 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 .
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