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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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Vector Space, HomK .V, ŒT C

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