Unformatted text preview: M we wrote the vectors in terms of the basis { v 1 ,v 2 } . In general, given a matrix of size n ⇥ n we can Fnd a special basis of R n so that it is easy to understand the behavior of M by using our special basis. This basis is special for the matrix M and does not work for all matrices. This is one motivation for introducing ‘coordinates’ with respect to an arbitrary basis. We start by seeing how to change coordinates fromone basis to another and then we learn how to express a matrix with respect to a new basis. Theorem 1 . Let V be a subspace of R n and let B = { v 1 ,v 2 , · · · v m } be a basis of V . Then we can write every element w in V uniquely in the form w = m X 1 a i v i , with a i 2 R . 1...
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 Fall '08
 MARKMAN
 Linear Algebra, Algebra

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