Math416_SimilarMatrices

Math416_SimilarMatrices - Math 416 - Abstract Linear...

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Unformatted text preview: Math 416 - Abstract Linear Algebra Fall 2011, section E1 Similar matrices 1 Change of basis Consider an n n matrix A and think of it as the standard representation of a transformation T A : R n R n . If we pick a different basis { v 1 ,...,v n } of R n , what matrix B represents T A with respect to that new basis? Write V = v 1 v 2 ... v n and consider the diagram R n standard T A A / R n standard id V- 1 $ H H H H H H H H H R n { v i } id V : v v v v v v v v v T A B = V- 1 AV / R n { v i } which says the new matrix is B = V- 1 AV . Remark: This is an instance of the more general change of coordinates formula. Start with a linear transformation T : V W . Let A be the old basis of V and e A the new basis. Let B be the old basis of W and e B the new basis. Then the diagram V A T [ T ] BA / W B id [id] e BB ! C C C C C C C C V e A id [id] A e A > } } } } } } } T [ T ] e B e A =[id] e BB [ T ] BA [id] A e A / W e B gives the change of coordinates formula [...
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Math416_SimilarMatrices - Math 416 - Abstract Linear...

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