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College Algebra Exam Review 174

# College Algebra Exam Review 174 - R W P 6 P 7(c Observe...

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184 3. PRODUCTS OF GROUPS 3.4.4. Using the previous exercise, identify . R 3 / ± with R 3 via the inner product h 2 4 ˛ 1 ˛ 2 ˛ 3 3 5 ; 2 4 ˇ 2 ˇ 2 ˇ 3 3 5 i D P 3 j D 1 ˛ j ˇ j . Given an ordered basis B D . v 1 ; v 2 ; v 3 / of R 3 , the dual basis B ± D ± v ± 1 ; v ± 2 ; v ± 3 ² of R 3 is deﬁned by the requirements h v i ; v ± j i D ı i;j , for 1 ± i;j ± 3 . Find the dual basis of B D 0 @ 2 4 1 2 1 3 5 ; 2 4 1 0 1 3 5 ; 2 4 0 1 1 3 5 1 A . 3.4.5. Give a different proof of Lemma 3.4.4 as follows: Let V be a ﬁnite dimensional vector space with ordered basis B D .v 1 ;v 2 ;:::;v n / . Let B ± D .v ± 1 ;v ± 2 ;:::;v ± n / be the dual basis of V ± . If v 2 V is nonzero, show that v ± j .v/ ¤ 0 for some j . 3.4.6. Prove parts (a) to (c) of Lemma 3.4.6 . 3.4.7. Let V be a ﬁnite dimensional vector space and let W be a subspace. Show that f 7! f j W is a linear map from V ± to W ± , and that the kernel of this map is W ı . 3.4.8. Let V be a ﬁnite dimensional vector space and let W be a subspace. Let ± W V ²! V=W be the quotient map. Show that g 7! ± ± .g/ D g ı ± is a linear isomorphism of .V=W / ± onto W ı . 3.4.9. Prove Corollary 3.4.9 . 3.4.10. Consider the R -vector space P n of polynomials of degree ± n with R -coefﬁcients, with the ordered basis ± 1;x;x 2 ;:::;x n ² . (a) Find the matrix of differentiation d dx W P 7 ! P 7 . (b) Find the matrix of integration
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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....
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