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e306fk - it in green ink on your Quiz. I used a scale of A...

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MAS 3105 Oct 16 , 2006 Quiz 3 and Key Prof. S. Hudson 1a) Find N ( A ), and 1b) Find a spanning set for N ( A ): A = ± 1 1 1 0 2 1 0 1 ² 2) Suppose that A and B are 3x3 matrices. Answer True or False: If A is row equivalent to B , then det A = det B . If B is row equivalent to a nonsingular matrix, then det ( B ) is nonzero. If det (2 A ) = 8 then det ( A - 1 ) = 1. If det ( A ) = 1 then A - 1 = adj A . The set of 2x2 matrices, denoted R 2 x 2 , is a vector space. 3) Choose ONE of these to prove. Assume A and B are nxn matrices. You can answer on the back. a) Prove that if U and V are subspaces of W then U V is one too. b) State and prove Cramer’s Rule. c) If S = { v 1 ,v 2 ...v n } , then span( S ) is a subspace of V [Mention all 4 parts of the definition of subspace, and prove parts 3 and 4 carefully]. Remarks and Answers: The quiz was given on Oct 16 (the original quiz had a typo - ‘Oct 12’). The average, among grades over 30, was about 40/60. The unofficial scale is: A’s = 49 to 60, B’s = 43 to 48, C’s = 37 to 42, D’s = 31 to 36, F’s = 00 to 30. Also, I added your 3 quiz scores to estimate your semester letter grade, and wrote
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Unformatted text preview: it in green ink on your Quiz. I used a scale of A = 80% and above, etc. [Of course, I will include your HW/etc into the calculation later on, and I will replace your lowest quiz grade with your MHW as promised - this method is just rough approximation. The scale may also change.] 1) The RREF of A is U = 1-1 1 1 2-1 The free variables are x 4 = and x 3 = . Get x 2 =-2 + and x 1 =- + . so, N ( A ) = { (- + ,-2 + , , ) T } . Find the spanning set by factoring out the Greek letters; S = { (1-2 1 0) T , (-1 1 0 1) T } . You should check that each vector in S satises Ax = 0 (because it is so easy). Other answers are possible. 2) FTTTT 3) From HW or text. Of course, you need to prepare for such proofs before each quiz. Id suggest practicing the explanations too - not just the calculations. 1...
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