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College Algebra Exam Review 170

# College Algebra Exam Review 170 - K with ordered bases B C...

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180 3. PRODUCTS OF GROUPS (b) Hom K .V; W / has dimension dim .V / dim .W / . Proof. The reader is invited to check that the map is linear. The map S C W W ! K n , which takes a vector in W to its coor- dinate vector with respect to C , is a linear isomorphism. For any T 2 Hom K .V; W / , the j –th column of OET Ł C;B is S C .T .v j // . If OET Ł C;B D 0 , then S C .T .v j // D 0 for all j and hence T .v j / D 0 for all j . It follows that T D 0 . This shows that T 7! OET Ł C;B is injective. Now let A D .a i;j / be any n –by– m matrix over K . We need to produce a linear map T 2 Hom K .V; W / such that OET Ł C;B D A . If such a T exists, then for each j , the coordinate vector of T .v j / with respect to C must be equal to the j –th column of A . Thus we require T .v j / D P n i D 1 a i;j w i WD a j . By Proposition 3.3.32 , there is a unique T 2 Hom K .V; W / such that T .v j / D a j for all j . This proves that T 7! OET Ł B;C is surjective. Assertion (b) is immediate from (a). n Proposition 3.4.11. Let V , W , X be finite–dimensional vector spaces over K with ordered bases
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Unformatted text preview: K with ordered bases B , C , and D . Let T 2 Hom K .V;W / and S 2 Hom K .W;X/ . Then ŒST Ł D;B D ŒSŁ D;C ŒT Ł C;B : Proof. Let B D .v 1 ;:::;v m / and C D .w 1 ;:::;w n / . Denote the dual basis of C by ± w ± 1 ;:::;w ± n ² . The j th column of ŒST Ł D;B is the coordinate vector with respect to the basis D of ST.v j / , namely S D .ST.v j // . The j th column of ŒSŁ D;C ŒT Ł C;B depends only on the j th column of ŒT Ł C;B , namely S C .T.v j // . The j th column of ŒSŁ D;C ŒT Ł C;B is ŒSŁ D;C S C .T.v j // D ³ S D .S.w 1 //;:::;S D .S.w n // ´ 2 6 4 h T.v j /;w ± 1 i : : : h T.v j /;w ± n i 3 7 5 D X k S D ı S.w k / h T.v j /;w ± k i D S D ı S X k w k h T.v j /;w ± k i ! D S D ı S.T v j /:...
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