082nd_tut6sol

082nd_tut6sol - MATH1111/2008-09/Tutorial VI Solution 1 Tutorial VI Suggested Solution 1 Let V be a vector space and B = b 1 ·· b n be an

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Unformatted text preview: MATH1111/2008-09/Tutorial VI Solution 1 Tutorial VI Suggested Solution 1. Let V be a vector space, and B = [ b 1 , ··· , b n ] be an ordered basis for V . (So dim V = n .) (a) Given any vectors u 1 , ··· , u m ∈ V and w ∈ V , show that w is a linear combination of u 1 , ··· , u m if and only if [ w ] B is a linear combination of the vectors [ u 1 ] B , ··· , [ u m ] B in R n . (b) Show that u 1 , ··· , u r ∈ V are linearly independent if and only if [ u 1 ] B , ··· , [ u r ] B are linearly independent vectors in R n . Ans . (a) It suffices to show that for any scalars α 1 , ··· ,α r , the coordinate vector of α 1 u 1 + α 2 u 2 + ··· + α r u r w.r.t. B is α 1 [ u 1 ] B + α 2 [ u 2 ] B + ··· + α r [ u r ] B . i.e. [ α 1 u 1 + α 2 u 2 + ··· + α r u r ] B = α 1 [ u 1 ] B + α 2 [ u 2 ] B + ··· + α r [ u r ] B . The details are left to you. (b) ” ⇒ ” part: Suppose α 1 [ u 1 ] B + α 2 [ u 2 ] B + ··· + α r [ u r ] B = (the zero vector in...
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This note was uploaded on 12/06/2010 for the course MATH MATH101 taught by Professor Chan during the Spring '09 term at HKUST.

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082nd_tut6sol - MATH1111/2008-09/Tutorial VI Solution 1 Tutorial VI Suggested Solution 1 Let V be a vector space and B = b 1 ·· b n be an

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