Math254Mid2Sols

# Math254Mid2Sols - MIDTERM 2 - SOLUTIONS MATH 254 - SUMMER...

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MIDTERM 2 - SOLUTIONS MATH 254 - SUMMER 2002 - KUNIYUKI CHAPTERS 4, 5 GRADED OUT OF 75 POINTS ¥ 2 = 150 POINTS TOTAL Assume that the "standard" operations for vector addition and scalar multiplication are being used in all relevant problems. 1) For each set below, circle "Yes" if it is a vector space, or circle "No" if it is not. Whenever you answer “Yes”, you do not have to justify your answer. • Whenever you answer “No”, give a counterexample. (10 points total) a) The set xy x y , () {} is an integer and is a real number (4 points) Yes No For example, (1, 0) is in the set, but p (1, 0) = ( , 0) is not, so the set is not closed under scalar multiplication. b) The set of all fourth-degree polynomials in x (4 points) Yes No For example, 0 is not in the set. c) Span , , 123 vvv , where v 1 , v 2 , and v 3 are vectors in R 6 (2 points) Yes No The span of a set of vectors in a vector space must be a subspace of the vector space.

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2) Let W be the set of all 33 ¥ real diagonal matrices. Prove that W is a subspace of M , . You may assume that M , is, itself, a vector space without having to prove that. (Completeness and quality are criteria for grading your proof.) (10 points) W is clearly a nonempty subset of M , . Let v and w be any two members of W . v = È Î Í Í Í ˘ ˚ ˙ ˙ ˙ v v v 1 2 3 00 , where the diagonal entries are real numbers. w = È Î Í Í Í ˘ ˚ ˙ ˙ ˙ w w w 1 2 3 , where the diagonal entries are real numbers. Prove closure of W under vector addition: vw += + + + È Î Í Í Í ˘ ˚ ˙ ˙ ˙ 11 22 . Note that the diagonal entries are real numbers. + is in W . Prove closure of W under scalar multiplication: Let c be any real scalar.
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## This note was uploaded on 09/08/2011 for the course MATH 254 taught by Professor Howard during the Spring '09 term at Mesa CC.

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Math254Mid2Sols - MIDTERM 2 - SOLUTIONS MATH 254 - SUMMER...

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