1.6 - Linear Algebra-115 Solutions to Fourth Homework...

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Unformatted text preview: Linear Algebra -115 Solutions to Fourth Homework Problem 2( d ) (Section 1 . 6) By Corollary 2 (page 47 in textbook) if a set is linearly independent and contains exactly n vectors, where n = dim( V ), then it is a basis. We know that dim( R 3 ) = 3. So, it suffices to check for linear independence. We form a matrix with these vectors as columns and we row reduce it: - 1 2- 3 3- 4 8 1- 3 2 ∼ - 1 2- 3 2- 1- 1- 1 ∼ - 1 2- 3- 1- 1 2- 1 ∼ - 1 2- 3 1 1- 3 ∼ - 1 2- 3 1 1 1 , which is an invertible matrix. Therefore, the three vectors are linearly indepen- dent. Problem 3( d ) (Section 1 . 6) Again dim( P 2 ( R )) = 3 and we will only check whether the polynomials are linearly independent or not. Assume a (4 x 2 + 2 x- 1) + b (- 10 x 2- 4 x + 3) + c (- 6 x 2- 5 x- 2) = 0 . Then, (4 a- 10 b- 6 c ) x 2 + (2 a- 4 b- 5 c ) x + (- a + 3 b- 2 c ) = 0 ,or 4 a- 10 b- 6 c 2 a- 4 b- 5 c- a + 3 b- 2 c =...
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This note was uploaded on 11/30/2009 for the course MATH 115A taught by Professor Liu during the Winter '07 term at UCLA.

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1.6 - Linear Algebra-115 Solutions to Fourth Homework...

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