# hw3sol - MATH 223 HOMEWORK#3 Solutions Problem 1 2(a,b,c in...

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MATH 223 - HOMEWORK #3 Solutions Problem 1. 2 (a,b,c) in Section 1.5. The set in (a) is linearly dependent because the second matrix is - 2 times the ﬁrst. In (b) there is no relation, so the set is independent. The set in (c) is again independent. Problem 2. 18 in Section 1.5. (You may assume that the set S is ﬁnite). Suppose we have a nontrivial relation among the polynomials: a 1 p 1 ( x ) + .. . + a n p n ( x ) = 0 . Let the polynomials be ordered so that deg p 1 < deg p 2 < ... < deg p n . Let m be the largest index such that a m 6 = 0, and let d be the degree of p m . The monomial x d occurs only in the polynomial p m because all the other p i with nonzero a i have smaller degree. It follows that a m x m = 0, hence a m = 0 and we have a contradiction. Problem 3. 9 in Section 1.6. Given ( a 1 ,a 2 ,a 3 ,a 4 ), we need to solve for b 1 ,b 2 ,b 3 ,b 4 such that ( a 1 ,a 2 ,a 3 ,a 4 ) = b 1 (1 , 1 , 1 , 1) + .. . + b 4 (0 , 0 , 0 , 1) . The unique solution is

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hw3sol - MATH 223 HOMEWORK#3 Solutions Problem 1 2(a,b,c in...

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