m235notes13-page5 - be a basis of R 2 . What is the matrix...

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We really should have subscripts that tell us what basis we are working with. We can just as well express the linear function f in terms of some other basis. Of course the entries in this new matrix will, in general, be di erent than the entries in the matrix arising from the standard basis. We now show how to compute the matrix of a linear function given its expression in terms of the standard basis in terms of an arbitrary basis. We actually do a little more than this. Example 5 . Let G be a linear transformation R 2 -! R 2 given by the matrix g = - 12 15 E E where E is the standard basis of R 2 .L e t A = f 1 = 3 2 ,f 2 = 4
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Unformatted text preview: be a basis of R 2 . What is the matrix of G wrt the basis A ? We Fnd the transition matrix from the A coordinates to the E coordinates. Note that we have f 1 = 1 A ! 3 2 E , f 2 = 1 A = 4 3 E . This says that the transition matrix from the basis A to the basis E is 1 E A = 3 4 2 3 . rom this we conclude that the transition matrix from the basis E to the basis A is 1 A E = (1 E A )-1 = 3-4-2 3 . The matrix for the linear map G in terms of the basis A is 1 A E g E E 1 E A . 5...
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This note was uploaded on 11/22/2011 for the course MATH 235 taught by Professor Markman during the Fall '08 term at UMass (Amherst).

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