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# Lecture6 - Lecture 6 Thm Let V V be finite-dimensional...

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1 Lecture 6 Thm: Let V V , be finite-dimensional vector spaces and V V T : is an isomorphism. If n b b b B ..., , , 2 1 is any basis of V, then ) ( ..., ), ( ), ( 2 1 n b T b T b T is a basis of V . 6.1 Matrix representation of transformation Review Section 2.3 Thm: Let T: R n → R m be a linear transformation. Then, there is an unique matrix A M m,n (R) such that x A x T ) ( for x R n where | | | ) ( ..., ), ( ), ( | | | 2 1 n e T e T e T A -- the standard matrix representation of T. and n e e e ..., , , 2 1 is the standard basis for R n Example: Let T: R 2 → R 3 be a linear transformation defined by 2 1 1 2 1 2 1 2 4 3 2 x x x x x x x T . Find the standard matrix representation A of T.

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2 Thm: Let V V , be finite-demensional vector spaces and, n b b b B ..., , , 2 1 and n b b b B ..., , , 2 1 be an ordered bases for V V , , respectively. Let V V T : be a linear transformation and m n R R T : be the linear transformation such that for each V v , we have B B v T v T ) ( ) ( . Then the standard matrix representation of T is the matrix A whose jth column vector is B i b T ) ( , and B B v A v T ) ( for all vectors V v .
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Lecture6 - Lecture 6 Thm Let V V be finite-dimensional...

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