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m235notes13-page4

# m235notes13-page4 - E coordinates to A coordinates We have...

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We can rewrite one of these equations as this as 0 @ 1 0 0 1 A B ! 0 @ a 1 a 2 a 3 1 A E . This means that we have made the coordinate change from the B coordinates to the E coordinates for the three vectors v 1 , v 2 , v 3 . From this we can write down the matrix that performs the coordinate change for all vectors expressed in terms of the basis B . It is 1 E B = 0 @ a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 1 A , Note that we have written out the algorithm for R 3 , but it works for R n and n. Example 4 . Let A = { v 1 = 1 - 1 , v 2 = 1 1 } be a bases of R 2 . We find the matrix which changes A coordinates into E coordinates. Since v 1 = 1 0 A -! 1 - 1 E , and, v 2 = 0 1 A -! 1 1 . we have that 1 E A = 1 1 - 1 1 . What is the matrix which transforms from
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Unformatted text preview: E coordinates to A coordinates. We have v E = 1 E A Â· v A (1 E A )-1 Â· v E = v A . We also have 1 A E Â· v E = v A . We conclude that 1 A E = (1 E A )-1 . Matrices and Change of Basis When we are given a linear function f : R m-! R n expressed as a matrix A , it has been understood, until now, that we are working with the standard basis of R n and the standard basis of R m . So an expression such as f : âœ“ x y â—†-! âœ“ 1-2 3 4 â—†âœ“ x y â—† 4...
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