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College Algebra Exam Review 164

College Algebra Exam Review 164 - /v ± j.v i D n X j D 1...

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174 3. PRODUCTS OF GROUPS for A; B; C 2 Hom K .V; W / and v 2 V . The reader is invited to check the remaining details in Exercise 3.4.1 . An important special instance of the preceeding construction is the vector space dual to V , Hom K .V; K/ , which we also denote by V . A linear map from V into the one dimensional vector space of scalars K is called a linear functional on V . V is the space of all linear functionals on V . Let us summarize our observations: Proposition 3.4.1. Let V be a vector space over a field K . (a) For any vector space W , Hom K .V; W / is a vector space. (b) In particular, V D Hom K .V; K/ is a vector space. Suppose now that V is finite dimensional with ordered basis B D .v 1 ; v 2 ; : : : ; v n / . Every element v 2 V has a unique expansion v D P n i D 1 ˛ i v i . For 1 j n define v j 2 V by v j . P n i D 1 ˛ i v i / D ˛ j . The functional v j is the unique element of V satisfying v j .v i / D ı i;j for 1 i n . 3 I claim that B D .v 1 ; v 2 ; : : : ; v n / is a a basis of V . In fact, for any f 2 V , consider the functional Q f D P n j D 1 f .v j /v j . We have Q f .v
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Unformatted text preview: /v ± j .v i / D n X j D 1 f.v j /ı i;j D f.v i /: Thus f.v i / D Q f .v i / for each element v i 2 B . It follows from Proposition 3.3.32 that f D Q f . This means that B ± spans V ± . Next we check the linear independence of B ± . Suppose P n j D 1 ˛ j v ± j D (the zero functional in V ± ). Applying both sides to a basis vector v i , we get D n X j D 1 ˛ j v ± j .v i / D n X j D 1 ˛ j ı i;j D ˛ i : Thus all the coefficients ˛ i are zero, which shows that B ± is linearly inde-pendent. B ± is called the basis of V ± dual to B . We showed above that for f 2 V ± , the expansion of f in terms of the basis B ± is f D n X j D 1 f.v j /v ± j : 3 Here ı i;j is the so called “Kronecker delta”, defined by ı i;j D 1 if i D j and ı i;j D otherwise....
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