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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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 Winter '07
 Liu
 Linear Algebra, Algebra, Vectors

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