hw3sol - MATH 223 - HOMEWORK #3 Solutions Problem 1. 2...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

hw3sol - MATH 223 - HOMEWORK #3 Solutions Problem 1. 2...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online