Lect25 - Chapter 4. Linear Transformations Math1111 Matrix...

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Unformatted text preview: Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Choices of Bases (Cont’d) Proof of Theorem. From the given condition, we have [ x ] E 2 = P [ x ] E 1 , for any x ∈ V , [ y ] F 2 = Q [ y ] F 1 , for any y ∈ W , [ L ( x )] F 1 = A [ x ] E 1 , for any x ∈ V . Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Choices of Bases (Cont’d) Proof of Theorem. From the given condition, we have [ x ] E 2 = P [ x ] E 1 , for any x ∈ V , [ y ] F 2 = Q [ y ] F 1 , for any y ∈ W , [ L ( x )] F 1 = A [ x ] E 1 , for any x ∈ V . Trace the diagram , we see that for any v ∈ V , [ L ( v )] F 1 = AP- 1 [ v ] E 2 , hence [ L ( v )] F 2 = QAP- 1 [ v ] E 2 Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Choices of Bases (Cont’d) Proof of Theorem. From the given condition, we have [ x ] E 2 = P [ x ] E 1 , for any x ∈ V , [ y ] F 2 = Q [ y ] F 1 , for any y ∈ W , [ L ( x )] F 1 = A [ x ] E 1 , for any x ∈ V . Trace the diagram , we see that for any v ∈ V , [ L ( v )] F 1 = AP- 1 [ v ] E 2 , hence [ L ( v )] F 2 = QAP- 1 [ v ] E 2 The matrix representation B of L r.t. E 2 and F 2 is QAP- 1 . Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Bases - Example Example . Let L : R 2 → R 3 be the linear transformation defined by L ( x ) = ( x 2 x 1 + x 2 x 1- x 2 ) T . Find the matrix representation of L relative to the ordered bases [ u 1 , u 2 ] and [ b 1 , b 2 , b 3 ] where u 1 = ( 1 2 ) T , u 2 = ( 3 1 ) T , and b 1 = ( 1 ) T , b 2 = ( 1 1 ) T , b 3 = ( 1 1 1 ) T . Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Bases - Example (Cont’d) Recall L : R 2 → R 3 , L ( x ) = ( x 2 x 1 + x 2 x 1- x 2 ) T . u 1 = ( 1 2 ) T , u 2 = ( 3 1 ) T , b 1 = ( 1 0 0 ) T , b 2 = ( 1 1 0 ) T , b 3 = ( 1 1 1 ) T . Ans . The standard matrix representation of L is 1 1 1 1- 1 Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Bases - Example (Cont’d) Recall L : R 2 → R 3 , L ( x ) = ( x 2 x 1 + x 2 x 1- x 2 ) T . u 1 = ( 1 2 ) T , u 2 = ( 3 1 ) T , b 1 = ( 1 0 0 ) T , b 2 = ( 1 1 0 ) T , b 3 = ( 1 1 1 ) T . Ans . The standard matrix representation of L is 1 1 1 1- 1 Let St 2 and St 3 denote the standard ordered bases for R 2 and R 3 resp.. Transition matrices: from [ u 1 , u 2 ] to St 2 : 1 3 2 1 ! ; from [ b 1 , b 2 , b 3 ] to St 3 : 1 1 1 1 1 1 . Chapter 4. Linear Transformations Math1111 Matrix Representations Change of Bases - Example (Cont’d) Recall L : R 2 → R 3 , L ( x ) = ( x 2 x 1 + x 2 x 1- x 2 ) T ....
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Lect25 - Chapter 4. Linear Transformations Math1111 Matrix...

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