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supplement_chap_4

supplement_chap_4 - Linear Algebra Fall...

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Linear Algebra, Fall 2007 (http://www.math.nthu.edu.tw/˜ wangwc/) Extra Materials for Chapter 4 This document is about how to obtain the matrix representation for the same linear transformation under different pairs of bases in the source and target spaces. Definition: Let x = x 1 x 2 . [ x ] 1 2 , 3 4 = α β if and only if x = α 1 2 + β 3 4 Thus α β is called the coordinate of x relative to the basis 1 2 , 3 4 Similarly, [ x ] 1 0 , 0 1 = x 1 x 2 An important identity: 1 3 2 4 α β = x 1 x 2 = α 1 2 + β 3 4 That is, 1 3 2 4 maps [ x ] 1 2 , 3 4 to [ x ] 1 0 , 0 1 via multiplication. In other words, 1 3 2 4 is the transition matrix from [ x ] 1 2 , 3 4 to [ x ] 1 0 , 0 1 . This is contrary to your first instinct. Make sure you understand it correctly. Find some examples from the textbook and exercises, or make up some examples and do the actual computation (change from one set of coordinate to another). Now Let L be a linear transformation from R 2 o R 3 . suppose that we know the matrix representation of L relative to standard basis is A : A = L 1 0 , 0 1 ,

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