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math 235 hw4

# math 235 hw4 - Homework 4 Due March 9 Adam Gamzon Related...

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Homework 4, Due March 9 Adam Gamzon Related reading: sections 3.3 and 3.4 from the text and the handout on dimension 1. Let T : R 3 R 3 be the linear transformation given by the matrix 5 - 4 - 2 - 4 5 - 2 - 2 - 2 8 . Let B = 2 2 1 , 1 - 1 0 , 0 1 - 2 . This is a basis of R 3 . (a) Compute the matrix [ T ] BB . Solution: We have 5 - 4 - 2 - 4 5 - 2 - 2 - 2 8 2 2 1 = 0 0 0 , 5 - 4 - 2 - 4 5 - 2 - 2 - 2 8 1 - 1 0 = 9 - 9 0 , 5 - 4 - 2 - 4 5 - 2 - 2 - 2 8 0 1 - 2 = 0 9 - 18 , so the coordinate vector of each of these with respect to B is 0 0 0 , 0 9 0 , 0 0 9 respectively. Hence [ T ] BB = 0 0 0 0 9 0 0 0 9 . (b) If x = 1 1 1 , find [ x ] B . Solution: Solving the system 1 1 1 = a 2 2 1 + b 1 - 1 0 + c 0 1 - 2 , we find a = 5 / 9 , b = 1

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- 1 / 9 and c = - 2 / 9. Hence [ x ] B = 5 / 9 - 1 / 9 - 2 / 9 .
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math 235 hw4 - Homework 4 Due March 9 Adam Gamzon Related...

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