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Unformatted text preview: ENSC 483 HW#2 SolutionM. Saif 1 Simon Fraser University School of Engineering Science ENSC 483  Modern Control Systems 1. Consider the following set of vectors: v 1 = 1 1 2 3 ; v 2 = 2 2 1 ; v 3 = 1 1 3 . 5 ; v 4 =  . 5 4 . 5 5 . 5 Define two basis B 1 = { v 1 , v 2 } , and B 2 = { v 3 , v 4 } for the subspace span of { v 1 , v 2 , v 3 , v 4 } . (a) Find the transformation matrix T , which transforms vectors represented in B 1 into vectors rep resented in B 2 basis. (b) Find the representation of the vector v = [1 3 4 2] T in the B 1 basis. (c) Use the results of a) to find the representation of v with respect to B 2 basis. (d) Verify that your answer in c) is correct. Solution: (a) Note that [ v 1 v 2 ] = [ v 3 v 4 ] T where T is a nonsingular 2 2 matrix that relates the two basis. Verify now that in this case: T = 4 9 2 1 2 1 2 1 (b) Note that v = [ v 1 v 2 ] where is the representation of v with respect to B 1 basis. You can easily calculate it to be = 1 1 (c) Based on the result in part (a), we can find the representation of v with respect to the B 2 basis as = T = 4 9 2 1 2 1 2 1 1 1 = 2 3 1 1 (d) To verify (c) simply try to directly find the representation of v with respect to B 2 : v = [ v 3 v 4 ] where is the representation of v with respect to B 2 basis. = 2 3 1 1 which agrees with the result in (c). ENSC 483 HW#2 SolutionM. Saif 2 2. Consider the following vectors in ( < 3 , < ) x 1 =  2 1 2 ; x 2 = 1 1 1 ; x 3 = 8 5 7 and following vectors in ( < 4 , < ): y 1 = 1 3 1 ; y 2 =...
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 Spring '07
 MehrdadSaif

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