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 .F r omth i sw ec anw r i t ed ownth em a t r ixth a t performs the coordinate change for all vectors expressed in terms of the basis B .I ti s 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 .W e±ndthema t r ixwh i chchange s A coordinates into E coordinates. Since v 1 = 1 0 A -! 1 - 1 E , and,
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Unformatted text preview: E coordinates to A coordinates. We have v E = 1 E A v A (1 E A )-1 v E = v A . We also have 1 A E 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 : x y -! 1-2 3 4 x y 4...
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