m235notes13-page1

m235notes13-page1 - M we wrote the vectors in terms of the...

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Notes 13: Coordinates Lecture Oct , 2011 Assume there are two supermarkets, one called A and the other B in a town of 1000 shoppers and that each month 1. 80% of the shoppers of A remain with A and the rest switch to B, and 2. 70% of the shoppers of B remain with B and the rest switch to A. The matrix M = . 8 . 3 . 2 . 7 describes the behavior of the shoppers in our town. If A ( n ) ,B ( n ) is the number of shoppers at A, B in month n, then . 8 . 3 . 2 . 7 ◆✓ A ( n ) B ( n ) = A ( n +1) B ( n +1) . We see that the vector v 1 = 600 400 is Fxed by M ,thatis , Mv 1 = v 1 . The vector v 2 = 500 - 500 has the property that Mv 2 = . 5 v 2 . If we write a vector w in terms of the basis v 1 ,v 2 ,i ti sea sytoseehow M,M 2 ,M 3 , ··· acts on w .I f w = a 1 v 1 + a 2 v 2 + v 2 ,then a 1 v 1 + a 2 v 2 ! a 1 v 1 + . 5 a 2 v 2 ! a 1 v 1 +( . 5 2 ) a 2 v 2 ! a 1 v 1 + a 2 ( . 5 3 ) v 2 ··· It is clear that in the long run the distribution of shoppers approaches the limit
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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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