The construction of the dual basis is a special case of the construction in the

The construction of the dual basis is a special case

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The construction of the dual basis is a special case of the construction in the proof of 2.4; take W = R with the single basis element w 1 = 1; then the v * i of (2.6.1.1) is equal to the f i 1 of (2.4.1). I-7
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Therefore a choice of basis { v i } yields an explicit isomorphism φ : V V * , namely φ ( v i ) = v * i (2 . 6 . 1 . 2) ((2.6.1.2) defines φ uniquely by (1.9).) Note that this isomorphism depends on the choice of basis; if we had chosen a different basis for V , we would have constructed a different isomorphism. There is no good way to select a “preferred” isomorphism from V to V * ; we will revisit this issue in Section 5 of this chapter. Remark 2.6.1.3 The element v * i of the dual basis depends not just on v i but on all the v j . If { w 1 , . . . , w n } is another basis and v 1 = w 1 , it does not follow that v * 1 = w * 1 . Definition 2.7. The double dual of V is the vector space V ** = ( V * ) * . 2.8. Applying (2.6) twice and invoking (1.4), we can construct an isomorphism V V ** . But there is a simpler, basis- independent construction. Map φ : V V ** by φ ( v )( g ) = g ( v ) and check that this map is one-one and onto. 2B. Multilinearity Definition 2.9. Let V , W and U be vector spaces. Then a bilinear map f : V × W U is a function satisfying the following axioms for v i V , w i W and α R : I-8
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i) f ( αv 1 + v 2 , w ) = αf ( v 1 , w ) + f ( v 2 , w ) ii) f ( v, αw 1 + w 2 ) = f ( v, w 1 ) + αf ( v, w 2 ) Remark 2.9.1. You can think of multilinearity as “linearity in each variable sepa- rately”. More precisely, we can restate conditions (2.9(i)) and (2.9(ii)) as follows: i 0 ) For each fixed v V , the map f v : W U defined by f v ( w ) = f ( v, w ) is linear. ii 0 ) For each fixed w W , the map f w : V U defined by f w ( v ) = f ( v, w ) is linear. Remark 2.2.1.2. As sets, V × W is the same thing as V W , so you might be tempted to think of a bilinear map as a function f : V W U But it is important to realize that this map is not in general a linear transformation. Here’s why: Applying axioms 2.9(i) and 2.9(ii), we find that for a bilinear map f we have f ( v 1 + v 2 , w 1 + w 2 ) = f ( v 1 , w 1 ) + f ( v 1 , w 2 ) + f ( v 2 , w 1 ) + f ( w 1 , w 2 ) whereas for a linear transformation f we have f ( v 1 + v 2 , w 1 + w 2 ) = f ( v 1 , w 1 ) + f ( v 2 , w 2 ) I-9
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which is not at all the same thing. We will want to adopt the convention that whenever we write down a map between vector spaces, it is assumed to be linear. Therefore, when f is a bilinear map, we will think of its domain as the set of ordered pairs V × W , rather than the vector space of ordered pairs V W . Definition 2.10. We generalize the notion of bilinearity as follows: A map f : V 1 × V 2 × · · · × V n W is called multilinear if it is linear in each variable separately. More precisely, the condition is that if we fix elements in n - 1 of the vector spaces V 1 , . . . V n , then the induced map on the remaining V i is linear. 2C. Tensor Products 2.11. The tensor product of two vector spaces V and W is a vector space V W which is the recipient of a “universal” multilinear map t : V × W V W (2 . 11 . 1) “Universal” means that any multilinear map with domain V × W can be factored uniquely as a composition g t where t : V W U is a linear map.
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