exam2SolutionsPctex

exam2SolutionsPctex - Math 222 - 200 Solutions Exam 2 April...

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Math 222 - 200 Solutions Exam 2 April 1, 2005 1. (30) Let T : V V be a linear transformation. Let { ~v 1 , 2 , 3 } be a basis of V . Suppose the matrix representation A of T with respect to this basis is A = 120 01 - 4 - 2 - 50 . (a) In terms of the basis vectors what does T ( 1 ) equal? The first column of A contains the coordinates of T ( 1 ), so T ( 1 )= ~ v 1 - 2 ~ v 3 (b) Does the equation T ( ~x ~ v 2 - ~ v 3 have a solution? If yes, find it, if no why not. In terms of coordinates the question is: does there exist an ~y R 3 such that A~y = 0 1 - 1 The augmented matrix for this system is 1200 - 41 - 2 - - 1 = 100 - 2 010 1 001 0 Thus, ~ y =[ - 2, 1, 0], and the solution to the original equation is = - 2 1 + 2 2. (50) A linear transformation E : V V is called a projection if it satisfies the equation E 2 = E . (a) Let V = R 3 .T h es e t U = { ~ u 1 , ~ u 2 , ~ u 3 } = {h 1, 1, 0 i , h 1, 0, 1 i , h 0, 1, 1 i} is a basis of R 3 . Define E by E ( λ 1 ~u 1 + λ 2 2 + λ 3 3 λ 1 ~ u 1 . Show that E is a projection. Note, this means you need to show that E
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exam2SolutionsPctex - Math 222 - 200 Solutions Exam 2 April...

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