082nd_tut4sol - MATH1111/2008-09/Tutorial IV Solution 1...

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MATH1111/2008-09/Tutorial IV Solution 1 Tutorial IV Suggested Solution 1. Consider the following vectors in R 4 : U = (2 1 - 1 1) T , V = ( - 1 1 1 1) T , W = ( - 4 1 3 1) T . (a) Which of the vectors X = (5 7 - 1 1) T , Y = (1 2 0 - 2) T is/are in Span( U, V, W )? (b) Are U, V, W linearly independent? Justify your answer. Ans . (a) To determine whether X or Y is in Span( U, V, W ), we need to check whether X or Y can be written as a linear combination aU + bV + cW of U, V, W where a, b, c are some scalars. Therefore, we consider the linear systems 2 - 1 - 4 1 1 1 - 1 1 3 1 1 1 a b c = 5 7 - 1 7 or 2 - 1 - 4 1 1 1 - 1 1 3 1 1 1 a b c = 1 2 0 - 2 . Applying elementary row operations, their augmented matrices are reduced to 1 0 - 1 0 1 2 0 0 0 0 0 0 4 3 0 0 or 1 0 - 1 0 1 2 0 0 0 0 0 0 0 0 1 0 . Hence we can solve for a, b, c in case of X but not Y . This implies X Span( U, V, W ) but Y / Span( U, V, W ). (b) We need to see: for what values of a, b, c will aU + bV + cW = 0 ? Consider the homogeneous system 2 - 1 - 4 1 1 1 - 1 1 3 1 1 1 a b c = 0 0 0 0 .
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