assign8_soln

# assign8_soln - Math 136 Assignment 8 Due: Wednesday, July...

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Math 136 Assignment 8 Due: Wednesday, July 14th 1. Let B = { ~ b 1 , ~ b 2 } and C = { ~ c 1 ,~ c 2 } be bases for R 2 . In each exercise, ﬁnd the change-of- basis matrix from B to C and the change-of-basis matrix from C to B . a) ~ b 1 = ± - 1 8 ² , ~ b 2 = ± 1 - 5 ² , ~ c 1 = ± 1 4 ² , ~ c 2 = ± 1 1 ² . Solution: To determine the change of coordinates matrix from B to C we need to calculate the C -coordinates of the images of the vectors in B . We get ± 1 1 - 1 1 4 1 8 - 5 ² ± 1 0 3 - 2 0 1 - 4 3 ² Hence, the change of coordinates matrix from B to C is C P B = ± 3 - 2 - 4 3 ² . To determine the change of coordinates matrix from C to B we need to calculate the B - coordinates of the images of the vectors in C . We get ± - 1 1 1 1 8 - 5 4 1 ² ± 1 0 3 2 0 1 4 3 ² Hence, the change of coordinates matrix from C to B is B P C = ( C P B ) - 1 = ± 3 2 4 3 ² . b) ~ b 1 = ± 7 - 2 ² , ~ b 2 = ± 2 - 1 ² , ~ c 1 = ± 4 1 ² , ~ c 2 = ± 5 2 ² . Solution: To determine the change of coordinates matrix from B to C we need to calculate the C -coordinates of the images of the vectors in B . We get ± 4 5 7 2 1 2 - 2 - 1 ² ± 1 0 8 3 0 1 - 5 - 2 ² Hence, the change of coordinates matrix from B to C is C P B = ± 8 3 - 5 - 2 ² . To determine the change of coordinates matrix from C to B we need to calculate the B - coordinates of the images of the vectors in C . We get ± 7 2 4 5 - 2 - 1 1 2 ² ± 1 0 2 3 0 1 - 5 - 8 ² Hence, the change of coordinates matrix from C to B is B P C = ( C P B ) - 1 = ± 2 3 - 5 - 8 ²

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2 2. For each of the following matrices, ﬁnd the inverse, or show that the matrix is not invertible. a) A = 1 2 0 2 2 5 1 - 1 3 . Solution: To determine if A is invertible we write [ A | I ] and row reduce: 1 2 0 1 0 0 2 2 5 0 1 0 1 - 1 3 0 0 1 1 0 0 11 / 9 - 2 / 3 10 / 9 0 1 0 - 1 / 9 1 / 3 - 5 / 9 0 0 1 - 4 / 9 1 / 3 - 2 / 9 Since the RREF of
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## This note was uploaded on 09/19/2011 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

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assign8_soln - Math 136 Assignment 8 Due: Wednesday, July...

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