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 andx 2 +1 ∈ S , but when we add these, we get x +1 6∈ S . This problem was from Ch 3.25. 3) Parts a) and c) are in the book and the lectures. Part b) is from the exercises. 1...
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 Spring '09
 JULIANEDWARDS
 Linear Algebra, Multiplication, Vector Space, col, Prof. S. Hudson

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