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Math254OldQuiz2Sols

# Math254OldQuiz2Sols - QUIZ 2 SOLUTIONS MATH 254 SUMMER 2001...

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QUIZ 2 - SOLUTIONS MATH 254 - SUMMER 2001 - KUNIYUKI CHAPTER 4 1) 3 4 3 3 0 2 4 4 2 1 9 0 6 16 8 4 9 16 0 8 6 4 25 8 10 v w - = - È Î Í Í Í ˘ ˚ ˙ ˙ ˙ - - È Î Í Í Í ˘ ˚ ˙ ˙ ˙ = - È Î Í Í Í ˘ ˚ ˙ ˙ ˙ - - È Î Í Í Í ˘ ˚ ˙ ˙ ˙ = - - ( ) - - - È Î Í Í Í ˘ ˚ ˙ ˙ ˙ = - - È Î Í Í Í ˘ ˚ ˙ ˙ ˙ 2) (Deleted) 3) The set (say, " W ") is a nonempty subset of R 5 ; 0 is guaranteed to be in it. Let x 1 and x 2 be any two members of W . Then, A x 0 1 = , and A x 0 2 = . Prove closure of W under vector addition: Show A x x 0 1 2 + ( ) = . A A A x x x x 0 0 0 1 2 1 2 + ( ) = + = + = So, x x 1 2 + is in W . Prove closure of W under scalar multiplication: Let c be any real scalar. Show A c x 0 1 ( ) = . A c c A c x x 0 0 1 1 ( ) = ( ) = ( ) = So, c x 1 is in W . Note: You could prove both types of closure simultaneously by showing A c x x 0 1 2 + ( ) = . Therefore, W is a subspace of R 5 .

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4) a) The set of all polynomials in x whose degree is exactly three Yes No Note: The answer would have been "Yes" if the set were "The set of all polynomials in x with degree three or less, including 0." b) The set of all real 4 3 ¥ matrices Yes No c) The set of standard basis vectors in R 5 Yes No For example, 0 is not a standard basis vector.
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Math254OldQuiz2Sols - QUIZ 2 SOLUTIONS MATH 254 SUMMER 2001...

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