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e511k - scored over 55 60 The scale for Q5 is A’s 46 to...

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MAS 3105 March 24, 2011 Quiz 5 and Key Prof. S. Hudson 1) Short Answers: a) [10pts; This is based on MHW 4.1]: Suppose W = triu(ones(2)), and we combine it’s columns into a basis F = { w 1 , w 2 } . Suppose L : R 2 R 2 and L ( w 2 ) = w 1 and L ( w 1 ) = 2 w 1 - w 2 . Find the matrix representation of L with respect to F . b) [5pts] What was the 2 × 2 matrix representation A in the rabbit story ? (recall each adult had two babies per year, which became adults in one year). c) [5pts] A transition matrix can be thought of as a matrix representation of which simple linear transformation ? 2) Let L : R 3 R 3 be L ( x ) = [ x 1 ,x 1 ,x 1 ] T . 2a) Find Ker(L). 2b) Find the range of L . 2c) Is L one-to-one ? (explain briefly) 2d) Is L onto R 3 ? (explain briefly) 3) Choose ONE of these. a) Suppose L : V W is linear. Show that ker ( L ) is a subspace of V . b) Suppose that A = ST and B = TS where S is nonsingular. Prove that A is similar to B . [This should be pretty short, mainly a calculation, but include some words, such as the definition of similar ]. c) Thm 3.6.6: Dim (Row ( A )) = Dim (Col( A )). Remarks and Answers: The Q5 average was about 37 / 60, rather low, but two students
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Unformatted text preview: scored over 55 / 60. The scale for Q5 is: A’s 46 to 60 B’s 40 to 45 C’s 34 to 39 D’s 28 to 33 F’s 0 to 27 I’ve updated your estimated semester grade, based on this quiz and on dropping your lowest quiz grade so far. See the upper right corner, in blue ink. I have still not included HW or MHW into the estimate. Since this is a new method compared to the Quiz 4 estimate, there might be a few surprises, but most of the grades stayed about the same. 1 1a) and 1b) M = ± 2 1-1 0 ² and A = ± 1 1 2 0 ² 1c) the identity transformation 2a) { e 2 , e 3 } 2b) { e 1 + e 2 + e 3 } 2c) No; the Ker contains non-zero vectors. 2d) No; the range (see 2b) is not R 3 . 3) See the text. I gave at least 15 points (usually much more) if the proof was logically correct (eg the steps could be justified), but only gave a perfect 20 if it included full explanations. 2...
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e511k - scored over 55 60 The scale for Q5 is A’s 46 to...

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