math110s-hw5sol - MATH 110: LINEAR ALGEBRA SPRING 2007/08...

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MATH 110: LINEAR ALGEBRA SPRING 2007/08 PROBLEM SET 5 SOLUTIONS 1. Let U,W,V be finite-dimensional vector spaces over F . Let α,β F . (a) Let T : V W be a linear transformation. Show that rank( T ) dim( V ) . Solution. Let dim( V ) = n . Suppose dim(im( T )) = rank( T ) > n . Then there must be at least n + 1 vectors w 1 ,..., w n +1 im( T ) that are linearly independent (eg. choose the first n + 1 vectors in a basis of im( T )). Since w 1 ,..., w n +1 are in im( T ), there exist v 1 ,..., v n +1 V such that T ( v i ) = w i , i = 1 ,...,n + 1. Since dim( V ) = n , v 1 ,..., v n +1 must be linearly dependent, so there exist α 1 ,...,α n +1 F , not all zero, such that α 1 v 1 + ··· + α n +1 v n +1 = 0 V . Hence we have found α 1 ,...,α n +1 F , not all zero, such that T ( α 1 v 1 + ··· + α n +1 v n +1 ) = T ( 0 V ) , ie. α 1 T ( v 1 ) + ··· + α n +1 T ( v n +1 ) = 0 W , ie. α 1 w 1 + ··· + α n +1 w n +1 = 0 W , which implies that w 1 ,..., w n +1 are linearly dependent — a contradiction. Hence our original assumption must have been false, ie. we must have dim(im( T )) n . (b) Let S : U V and T : V W be linear transformations. Show that rank( T ◦ S ) rank( T ) and rank( T ◦ S ) rank( S ) . Solution. Recall the following elementary fact from set theory: A B implies f ( A ) f ( B ) for any function f . In particular, since S ( U ) V , we must have T ( S ( U )) ⊆ T ( V ), ie. T ◦ S ( U ) ⊆ T ( V ), ie. im( T ◦ S ) im( T ) (recall that im( ϕ ) is just another way of writing ϕ ( V ) — both denote the range of ϕ ). Now both of these are subspaces of W and so dim(im( T ◦ S )) dim(im( T )) , ie. rank( T ◦ S ) rank( T ) . For the second part, we define the function ϕ : im( S ) W by ϕ ( v ) = T ( v ) for all v im( S ). Note that ϕ is a linear transformation since for all v 1 , v 2 im( S ) and α,β F , we have ϕ ( α v 1 + β v 2 ) = T ( α v 1 + β v 2 ) = α T ( v 1 )+ β T ( v 2 ) = αϕ ( v 1 )+ βϕ ( v 2 ). So applying part (a) to ϕ , we get rank( ϕ ) dim(im( S )) = rank( S ) . (1.1) Date : May 3, 2008 (Version 1.0). 1
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But note that im( T ◦ S ) im( ϕ ) — since if w im( T ◦ S ), then w = T ( S ( u )) for some u U and therefore w = T ( v ) = ϕ ( v ) where v = S ( u ) im( S ); hence w im( ϕ ). So dim(im( T ◦ S )) dim(im( ϕ )) , ie. rank(
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math110s-hw5sol - MATH 110: LINEAR ALGEBRA SPRING 2007/08...

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