e483h2s_07

# e483h2s_07 - ENSC 483 HW#2 Solution–M Saif 1 Simon Fraser...

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Unformatted text preview: ENSC 483 HW#2 Solution–M. 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 ] β where β is the representation of v with respect to B 1 basis. You can easily calculate it to be β = • 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 ˆ β = Tβ = 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 ] ˆ β where ˆ β is the representation of v with respect to B 2 basis. ˆ β = 2 3 • 1- 1 ‚ which agrees with the result in (c). ENSC 483 HW#2 Solution–M. 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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e483h2s_07 - ENSC 483 HW#2 Solution–M Saif 1 Simon Fraser...

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