e506k - S is a subspace of V Prove that L S is a subspace...

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MAS 3105 June 8, 2006 Quiz 5 Key Prof. S. Hudson 1) Answer TRUE or FALSE: a) If A and B are row equivalent, then Col( A T )=Col( B T ). b) If A is similar to a nonsingular matrix, then A is nonsingular. c) If A is similar to I , then A = I . d) If A and B are row equivalent they have the same rank. e) If A represents L : R 3 R 2 then Col ( A ) = L ( R 3 ). 2) Find the transition matrix representing the change in coordinates from [1,x] to [2x- 1,2x+1] in P 2 . 3) Choose ONE of these. a) Suppose A and B are nxn and B is nonsingular. Prove that rank ( BA ) = rank ( A ). [Hint: compare their nullspaces] b) Suppose that L : V W is linear and
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Unformatted text preview: 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 ). [Explain each step, including a couple of sentences about dependency relations]. Remarks and Answers: The average was about 45/60. 1) TTTTT 2) The matrix U that goes back to the standard basis is easier to do 1st; then find its inverse any way you choose (I used the adjoint method). U = ±-1 1 2 2 ² so , the answer is U-1 = 1-4 ± 2-1-2-1 ² 3) See text. 1...
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