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e509k

# e509k - four quiz grades so far The average for that is...

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MAS 3105 June 4, 2009 Quiz 5 and Key Prof. S. Hudson 1) [30pts]These two matrices are row equivalent. U is in REF but not RREF. A = 5 0 5 0 3 2 5 6 2 0 2 3 U = 1 0 1 7 0 1 1 3 0 0 0 1 a) Find a dependency relation for the columns of A . b) Find a basis of col ( A ). c) Find a basis of N ( A ) and ﬁnd the nullity of A . 2) [10pts] Find the transition matrix from the standard basis of R 2 to the basis B = { [0 , 1] T , [2 , 0] T } . Circle your answer. 3) Circle ONE of these proofs, and answer on the back. a) Let L : V W be linear. Prove that Ker(L) is a subspace of V . b) If Ker ( L ) = { 0 } then L is 1-1. c) Thm 3.6.6: Dim (Row ( A )) = Dim (Col( A )). Remarks and Answers: The average was approx 47 out of 60, which is pretty good, especially for Chapter 3 material. Many people didn’t seem to know exactly what a dependency relation is (1a), but the results on the other problems were good. The scale is A’s = 52-60, B’s = 46-51, C’s= 40-45, etc. My new estimate for your semester grade is in the upper right. It is based on your best
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Unformatted text preview: four quiz grades, so far. The average for that is approx 194 out of 240. 1a) a 1 + a 2-a 3 + 0 a 4 = . A dependency relation is an equation that shows a set of vectors ﬁts the deﬁnition of LD. Ideally, it should contain all 4 vectors and should have the zero vector on the RHS. 1b) Use the leading ones in U to choose { a 1 , a 2 , a 4 } . Actually, any basis of R 3 is also OK, but I didn’t give full credit for other bases without some valid explanation or work. 1c) A basis is [1 , 1 ,-1 , 0] T (essentially the same as the answer to 1a). Nullity = 1. 2) Combine the vectors into a matrix B and ﬁnd its inverse. It has zeroes on the diagonal and b 12 = 1 and b 21 = 1 / 2. It is always a good idea to check your answer to any B-1 calculation, since it only takes a few seconds. I usually give less partial credit in this situation. 3) See text or HW. 1...
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