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e405k

# e405k - ⊂ V then span B is a subspace of V Answers The...

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MAS 3105 June 3, 2005 Quiz 4 Key Prof. S. Hudson Show all your work and reasoning for maximum credit. If you continue your work on another page, be sure to leave a note. Do not use a calculator, book, or any personal paper. You may ask about any ambiguous questions or for extra paper. If you use extra paper, hand it in with your exam. 1) The matrix A is row equivalent to U . a) Find a basis for N ( A ), and explain brieﬂy how you know it is a basis. b) Do the same for Col A . A = 1 3 0 9 4 1 1 4 19 2 1 1 2 11 2 U = 1 0 0 0 1 0 1 0 3 1 0 0 1 4 0 2) Are the following sets subspaces of P 4 ? Be careful ! And explain each answer brieﬂy. a) The polynomials p ( x ) in P 4 such that p (0) = 0. b) The polynomials in P 4 of even degree. 3) Choose ONE of these to prove. You can answer on the back. a) If dim(V)= n > 0 and B = { v 1 ,v 2 ,...,v n } is L.I., then B is a basis of V. b) If U and V are subspaces of W then so is U V . c) If B = { v 1 ,v 2 ,...,v n
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Unformatted text preview: } ⊂ V , then span ( B ) is a subspace of V . Answers: The average grade this time was barely over 30/60, with fairly poor average results on all three problems. 1a) See Spring 2004, Quiz 4. 1b) The leading ones in U appear in columns 1, 2 and 3. So, the ﬁrst three columns of A are a basis for Col ( A ). 2a) Yes. It is closed under addition; if p and q are in the set S , then p (0) = 0 and q (0) = 0. So, ( p + q )(0) = p (0) + q (0) = 0 and p + q ∈ S . A similar calculation shows S is closed under scalar multiplication. 2b) No. For example, x 2 + x ∈ S and-x 2 +1 ∈ S , but when we add these, we get x +1 6∈ S . This problem was from Ch 3.2-5. 3) Parts a) and c) are in the book and the lectures. Part b) is from the exercises. 1...
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