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e411k - MAS 3105 Quiz 4 and Key March 3 2011 Prof S Hudson...

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MAS 3105 March 3, 2011 Quiz 4 and Key Prof. S. Hudson 1a) Let u 1 = [1 , 2] T and u 2 = [0 , 1] T . Let v 1 = [2 , 4] T and v 2 = [1 , 1] T . Find the transition matrix from { v 1 , v 2 } to { u 1 , u 2 } . 1b) Use this to write 2 v 1 + v 2 as a linear combination of u 1 and u 2 . 2) True-False: P 5 is a subspace of C [0 , 1]. If A is a singular 3x3 matrix, then its nullity is at least 1. If L is a list of n vectors, then dim (span L) = n. is a subspace of R 2 × 3 . If L is a list of 3 L.I. functions in V , then dim V 3. 3) Choose ONE (L.I. means linearly independent). You can answer on the back. a) Prove this part of the 2/3 thm: If L spans V with exactly n vectors, and dim V = n , then L is L.I. b) [3.3.2] ]Let L be L.I. in V , and v span ( L ). Prove v can be written uniquely as a linear combination of vectors in L . c) Suppose A is a nonsingular 3x3 matrix. Prove that its columns are L.I. in R 3 . Remarks and Answers: The average was approx 42 / 60. The scale is A’s 50-60 B’s 44-49 C’s 38-43 D’s 32-37 F’s 00-31 I estimated your semester grade by averaging your 4 quiz grades; see the upper right corner of your quiz. This estimate is probably accurate for students with no extreme
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