e409k - MAS 3105 Quiz 4 and Key Prof S Hudson 1 Let x 1 =(1...

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Unformatted text preview: MAS 3105 May 28, 2009 Quiz 4 and Key Prof. S. Hudson 1) Let x 1 = (1 , 1 , 1) T and x 2 = (3 ,- 1 , 4) T . Find a third vector x 3 so that { x 1 , x 2 , x 3 } is a basis of R 3 . Explain briefly how you know this is a basis. 2) Which of the following are subspaces of R 3 ? Explain each answer briefly. a) S = { ( x 1 ,x 2 ,x 3 ) T | 3 x 1 + x 3 = 0 } b) S = { ( x 1 ,x 2 ,x 3 ) T | x 1 x 3 = x 2 } c) S = { ( x 1 ,x 2 ,x 3 ) T | x 1 = 5 x 2 = x 3 } d) S = { ( x 1 ,x 2 ,x 3 ) T | | x 1 | + | x 2 | = 1 } 3) Choose ONE a) Prove this part of the 2/3 thm: If L is a L.I. set of n vectors in V , and dim V = n , then L spans V . b) Let L be a finite list of vectors in V . Prove that span ( L ) is a subspace of V . To save time, you can skip the parts about L ⊂ V and about closure under addition. But prove the other two parts. c) Suppose A is a nonsingular 3x3 matrix. Prove that its columns span R 3 . Include the definition of span (don’t take a shortcut such as saying that the columns are actually a...
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e409k - MAS 3105 Quiz 4 and Key Prof S Hudson 1 Let x 1 =(1...

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