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# e504k - b Suppose that L V → W is linear and S is a...

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MAS 3105 Feb 10, 2004 Quiz 5 and Key Prof. S. Hudson 1) Answer TRUE or FALSE: a) If A is similar to B then A - 4 I is similar to B - 4 I . b) If A is similar to a singular matrix then A is also singular. c) If A represents a 60 rotation of R 2 then A 6 = I . d) If A and B are row equivalent, they have the same column space. e) If A represents L : R 3 R 2 then Col ( A ) = L ( R 3 ). 2) Find the transition matrix from [ v 1 , v 2 , v 3 ] to [ u 1 , u 2 , u 3 ] where: v 1 = (4 , 6 , 7) T , v 2 = (0 , 1 , 1) T , v 3 = (0 , 1 , 2) T u 1 = (1 , 1 , 1) T , u 2 = (1 , 2 , 2) T , u 3 = (2 , 3 , 4) T 3) Choose ONE of these to prove (on the back of the page). a) If A and B are row equivalent, they have the same row space.
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Unformatted text preview: b) Suppose that L : V → W is linear and S is a subspace of V . Prove that L ( S ) is a subspace of W . c) The dimension of Col(A) equals the dimension of Row(A). Answers: 1) TTTFT 2) This was a HW problem. Set S = U-1 V (maybe from an arrow diagram), which is easiest to get from [ U | V ] → [ I | S ]. S = 1-1-2 1 1 1 1 3) These proofs are in the textbook [though I think my proof of c) is clearer than the book’s]. 1...
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