College Algebra Exam Review 161

College Algebra Exam Review 161 - 3.3. VECTOR SPACES 171...

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Unformatted text preview: 3.3. VECTOR SPACES 171 Proof. First, let’s check that W is finite–dimensional, with dimension no greater than dim.V /. If fv1 ; : : : ; vn g is a basis of V , then fT .v1 /; : : : ; T .vn /g is a spanning subset of W , so contains a basis of W as a subset. Now let fw1 ; : : : ; ws g be a basis of W . For each basis element wi , let xi be a preimage of wi in V (i.e., choose xi such that T .xi / D wi ). The map wi 7! xi extends uniquely to a linear map S W W ! V , defined P P by S. P ˛i wi / D ˛i i i P xi , according to Proposition 3.3.32. We have P P T ı S. i ˛i wi / D T . i ˛i xi / D i ˛i T .xi / D i ˛i wi . Thus T ı S D idW . I In the situation of the previous proposition, let W 0 denote the image of S . I claim that V D ker.T / ˚ W 0 Š ker.T / ˚ W: Suppose v 2 ker.T / \ W 0 . Since v 2 W 0 , there is a w 2 W such that v D S.w/. But then 0 D T .v/ D T .S.w// D w , and, therefore, v D S.w/ D S.0/ D 0. This shows that ker.T / \ W 0 D f0g. For any v 2 V , we can write v D S ı T .v/ C .v S ı T .v//. The first summand is evidently in W 0 , and the second is in the kernel of T , as T .v/ D T ı S ı T .v/. This shows that ker.T / C W 0 D V . We have shown that V D ker.T / ˚ W 0 . Finally, note that S is an isomorphism of W onto W 0 , so we also have V Š ker.T / ˚ W . We have shown the following: Proposition 3.3.35. If T W V ! W is a linear map and V is finite– dimensional, then V Š ker.T / ˚ range.T /. In particular, dim.V / D dim.ker.T // C dim.range.T //. Now let V be a finite–dimensional vector space and let N be a subspace. The quotient map W V ! V =N is a a surjective linear map with kernel N . Let S be a right inverse of , as in the proposition, and let M be the image of S . The preceding discussion shows that V D N ˚ M Š N ˚ V =N . We have proved the following: Proposition 3.3.36. Let V be a finite–dimensional vector space and let N be a subspace. Then V Š N ˚ V =N . In particular, dim.V / D dim.N / C dim.V =N /. Corollary 3.3.37. Let V be a finite–dimensional vector space and let N be a subspace. Then there exists a subspace M of V such that V D N ˚ M . ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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