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# Lecture11 - .1 Similarity of representations relative to...

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1 Lecture 11 11.1 Similarity of representations relative to different bases (Cont.) Let n n R R T : be a linear transformation. To find the relationship of the matrix representations relative to different bases: T R n R n B B R B Change basis B B C , B B C , T R n R n B B R B Remark: 1) There is an invertible matrix C s. t. C R C R B B 1 . 2) B R and B R are similar matrices. 3) Two n n matrices are similar if they are matrix representations of the same linear transformation T relative to suitable ordered bases. Example: Let T: R 3 → R 3 by T(x 1 , x 2 , x 3 ) = (x 1 + x 2 + x 3 , x 1 + x 2 , x 3 ). 1) Find standard matrix representation A of T. 2) Find matrix representation B R of T relative to B where 1 1 0 , 1 0 1 , 0 1 1 B . 3) Find invertible matrix C, s.t. AC C R B 1 .

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2 Example: Let T : P 2 → P 2 be the linear transformation defined by ) ( )) ( ( x p x x p T and let B 1 = {1, x, x 2 } and B 2 = {x, x +1, x 2 – 1} be two ordered basis of P 2 . Find the matrix representations R B 1 , R B 2 of T and a matrix C s. t. R B 2 = C -1 R B 1 C. 11.2 The interplay of matrices and linear transformation Let V be an n-dimensional vector space and T: V → V be a linear transformation. If there is a 0 v V, s. t. v v T ) ( , Let B and B be two distinct ordered bases of V and the matrix representations of T relative to B and B be R B and B R , respectively. Then B B B v v R and B B B v v R Thm: Let A and R be similar n n matrices. i.e. there is an invertible C, s. t, R = C
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Lecture11 - .1 Similarity of representations relative to...

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