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Bilinear Maps and Tensor Products

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Bilinear Maps and Tensor Products Extra credit 6 Math 25a, Fall 2018 Bilinear maps and tensor products This assignment is extra credit and is due Friday, November 9. You may use any resource. Bilinear maps. Let V; W; U be vector spaces over F . A function φ : V × W ! U is called bilinear if φ(v + v0; w) = φ(v; w) + φ(v0; w) and φ(c v; w) = c φ(v; w) φ(v; w + w0) = φ(v; w) + φ(v; w0) and φ(v; c w) = c φ(v; w): Another way to say this is that φ is linear in each coordinate separately, i.e. if we fix a vector w 2 W , then v 7! φ(v; w) defines a linear map φ(·; w): V ! U, and similarly, for v 2 V , the map φ(v; ·): W ! U is linear. Example. A bilinear map that you may have seen before and that we'll study later is the dot n n product R × R ! R, which is defined as v · w = v1w1 + ··· + vnwn. a b Example. For every symmetric matrix A = , there is a bilinear map φ : 2 × 2 ! b c A R R R defined by t φA(v; w) = v Aw: How is the dot product related to this construction? Let B(V × W; U) be the set of bilinear maps V × W ! U. Give B(V × W; U) the structure of a vector space. Assume V; W have bases v1; : : : ; vn and w1; : : : ; wm. As we did for linear maps, observe that (1) φ 2 B(V ×W; U) is determined by its values on pairs of basis elements (vi; wj), (2) if φ, 2 B(V ×W; U) Extra credit 6 Math 25a, Fall 2018 agree on (vi; wj) for 1 ≤ i ≤ n and 1 ≤ j ≤ m, then φ = , and (3) given uij 2 U for 1 ≤ i ≤ n and 1 ≤ j ≤ m, there is a bilinear map φ : V × W ! U such that φ(vi; wj) = uij. Assume V; W; U are all finite dimensional. Determine dim B(V × W; U) by finding a basis. Extra credit 6 Math 25a, Fall 2018 Tensor product of vector spaces. Let V and W be vector spaces over F . We're going to construct a new vector space called the tensor product V ⊗ W , and we'll see that this vector space is useful for a variety of reasons. The vector space V ⊗ W is defined using quotient spaces.1 First let E be the vector space that is spanned by linearly independent vectors fδv;w : v 2 V; w 2 W g. In other words, a vector in E looks like c1 δv1;w1 + ··· + ck δvk;wk ; 0 where v1; : : : ; vk 2 V and w1; : : : ; wk 2 W are any vectors. Let E ⊂ E be the subspace spanned by all the vectors of the following forms δv+v0;w − δv;w − δv0;w δv;w+w0 − δv;w − δv;w0 δcv;w − c δv;w δv;cw − c δv;w 0 Define V ⊗ W = E=E . We denote the equivalence class [δv;w] by v ⊗ w. There is an obvious map i : V × W ! V ⊗ W defined by i(v; w) = v ⊗ w. Is this map linear/bilinear/neither? An element of Im(i) is called a pure tensor. Explain why V ⊗ W is spanned by pure tensors, but not every vector in V ⊗ W is a pure tensor if dim V ≥ 2 and dim W ≥ 2. 1If you haven't done the quotient space extra credit, go back and do it. You can turn it in with this assignment for credit. Extra credit 6 Math 25a, Fall 2018 Let φ : V × W ! U be a bilinear map. Show that there is a unique linear map Φ : V ⊗ W ! U such that Φ ◦ i = φ. Prove that the association φ 7! Φ defines a linear isomorphism B(V × W; U) !L(V ⊗ W; U). Use the above to prove that if v1; : : : ; vn and w1; : : : ; wm are bases for V; W , then fvi ⊗ wj : 1 ≤ i ≤ n; 1 ≤ j ≤ mg is a basis for V ⊗ W . Use the previous problem to conclude that V ⊗ F is isomorphic to V . Extra credit 6 Math 25a, Fall 2018 Complexification. Let F and K be fields such that F ⊂ K (a good example to think about is 2 F = R and K = C). Observe that K is a vector space over F . Then for any vector space V over F , we have K ⊗ V , which is a vector space over F . Check that K ⊗ V has the structure of a vector space over K, where the scalar multiplication is c(k ⊗ v) = (ck) ⊗ v. Show that if dimF V = n, then also dimK (K ⊗ V ) = n. When F = R and K = C, the assignment V 7! C ⊗ V associates to every real vector space a complex vector space of the same dimension. This procedure is called complexification or sometimes extension of scalars. 2 The case F = R and K = C was on HW3..
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