Unformatted text preview: R W P 6 ! P 7 . (c) Observe that multiplication by 1 C 3x C 2x 2 is linear from P 5 to P 7 , and ﬁnd the matrix of this linear map. 3.4.11. Let B D .v 1 ;:::;v n / be an ordered basis of a vector space V over a ﬁeld K . Denote the dual basis of V ± by B ± D ± v ± 1 ;:::;v ± n ² . Show that for any v 2 V and f 2 V ± , h v;f i D n X j D 1 h v;v ± j ih v j ;f i : 3.4.12. Let B D .v 1 ;:::;v n / and C D .w 1 ;:::;w n / be two bases of a vector space V over a ﬁeld K . Denote the dual bases of V ± by B ± D ± v ± 1 ;:::;v ± n ² and C ± D ± w ± 1 ;:::;w ± n ² . Recall that Œ id Ł B;C is the matrix with .i;j/ entry equal to h w j ;v ± i i , and similarly, Œ id Ł C;B is the matrix with .i;j/ entry equal to h v j ;w ± i i . Use the previous exercise to show that Œ id Ł B;C and Œ id Ł C;B are inverse matrices....
View
Full Document
 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Vector Space, WI, hvj, @425, W ı

Click to edit the document details