081st_tut5sol

081st_tut5sol - MATH1111/2008-09/Tutorial V Solution 1...

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Unformatted text preview: MATH1111/2008-09/Tutorial V Solution 1 Tutorial V 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- 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- 2     . Applying elementary row operations, their augmented matrices are reduced to     1 0- 1 0 1 2 0 0 0 0 4 3     or     1 0- 1 0 1 2 0 0 0 0 1     . 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 = ?...
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This note was uploaded on 09/06/2010 for the course MATH MATH1111 taught by Professor Forgot during the Fall '08 term at HKU.

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081st_tut5sol - MATH1111/2008-09/Tutorial V Solution 1...

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