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 ∈ W , then v 7→ φ(v, w) defines a linear map φ(·, w): V → U, and similarly, for v ∈ 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 ∈ B(V ×W, U) is determined by its values on pairs of basis elements (vi, wj), (2) if φ, ψ ∈ 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 = ψ, and (3) given uij ∈ 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 {δv,w : v ∈ V, w ∈ W }. In other words, a vector in E looks like

c1 δv1,w1 + ··· + ck δvk,wk , 0 where v1, . . . , vk ∈ V and w1, . . . , wk ∈ 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

{vi ⊗ wj : 1 ≤ i ≤ n, 1 ≤ j ≤ m} 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.