081st_tut7sol

081st_tut7sol - MATH1111/2008-09/Tutorial VII Solution 1...

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MATH1111/2008-09/Tutorial VII Solution 1 Tutorial VII Suggested Solution 1. Let V be a (general) vector space, and B = { b 1 , ··· , b n } be a basis for V . (So dim V = n .) (a) Show that u = w if [ u ] B = [ w ] B (i.e. the coordinate mapping from V to R n is one-to-one). (b) Show that for any ( y 1 y 2 ··· y n ) T R n , there exists a vector u V such that [ u ] B = ( y 1 y 2 ··· y n ) T (i.e. the coordinate mapping is onto). (c) 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 . (d) 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) Let [ u ] B = [ w ] B = ( a 1 a 2 ··· a n ) T . Then by the deﬁnition of coordinates, both u and w = a 1 b 1 + a 2 b 2 + ··· + a n b n . So

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081st_tut7sol - MATH1111/2008-09/Tutorial VII Solution 1...

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