082nd_tut5sol

# 082nd_tut5sol - MATH1111/2008-09/Tutorial V Solution 1...

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MATH1111/2008-09/Tutorial V Solution 1 Tutorial V Suggested Solution 1. Let V be a vector space and dim V = 2009, and let S, T be subspaces of V . Prove or disprove the following. (a) For every integer r with 0 r 2009, one can always find a subspace W r of V such that dim W r = r . (b) S = T if and only if dim S = dim T . (c) dim( S T ) min(dim S, dim T ) where min( a, b ) = the minimum of a and b . (d) dim( S T ) max(dim S, dim T ) where max( a, b ) = the maximum of a and b . (e) Let v V and U = Span( v ). Then dim( U + S ) = dim S + 1. Ans . Let { v 1 , v 2 , · · · , v 2009 } be a basis for V (since dim V = 2009). (a) True. For r = 0, we take W 0 = { 0 } which is a subspace of V and dim W 0 = 0. For 1 r 2009, we take W r to be the subspace Span( v 1 , v 2 , · · · , v r ). You can check easily that v 1 , v 2 , · · · , v r forms a basis for W r . So dim W r = r . (b) False. The ”only if” part is obviously correct (i.e. S = T are subspaces dim S = dim T .) However the ”if” part is wrong. Consider V = R 2 and let S = Span( e 1 ) and T = Span( e 2 ).

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