e505k - If L V → W is linear and T is a subspace of W...

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MAS 3105 June 9, 2005 Quiz 5 and Key Prof. S. Hudson Name Show all your work and reasoning for maximum credit. If you continue your work on another page, be sure to leave a note. Do not use a calculator, book, or any personal paper. You may ask about any ambiguous questions or for extra paper. If you use extra paper, hand it in with your exam. 1) [20 pts] Answer TRUE or FALSE: Two row equivalent matrices must have the same column space. If A is similar to A 2 , then A 2 is similar to A 4 . If A represents L : R 3 R 2 then A is a 2x3 matrix. If A is an mxn matrix then A and A T have the same rank. If A is an mxn matrix then A and A T have the same nullity. 2) [20 pts] Let L : R 2 R 2 be a linear transformation such that L ((1 , 2) T ) = (2 , 3) T and L ((2 , 3) T ) = (4 , 5) T . Find L ((3 , 4) T ). 3) [20 pts] Choose ONE of these to prove. a) Edited 6/13/05
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Unformatted text preview: : If L : V → W is linear, and T is a subspace of W , then L-1 ( T ) is a subspace of V . b) Prove that dim(Row( A )) = dim(Col( A )). [If you follow my outline, explain each step and include the phrase ”dependency relation” when it is needed]. c) Show that if A is similar to B then A T is similar to B T . Answers: 1) FTTTF 2) [6 7] T . I used [3 4] T = 2[2 3] T-[1 2] T to get this, but there are other ways (you could find the matrix rep for L instead, for example). 3) a) = HW and b) = Thm. The easiest is part c): Since A = S-1 BS , we can transpose both sides and get A T = S T B T ( S-1 ) T . But ( S-1 ) T = ( S T )-1 , by previous HW, so A T is similar to B T . 1...
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