1. Tensor product
Let V, W be finite dimensional vector spaces over a field K. Given v ∈ V and w ∈ W, we define a function v ⊗ w : V∗× W∗ → K by
(v ⊗ w)(ϕ, ψ) = ϕ(v)ψ(w).
Then v ⊗ w is a K-bilinear map from V∗× W∗ to K. The set of all K-bilinear maps from V∗× W∗ to K is denoted by V ⊗KW and called the tensor product of V and W. The space V ⊗KW is spanned by {v ⊗ w : v ∈ V, w ∈ W }.
Lemma 1.1. Let α = {vi : 1 ≤ i ≤ n} and β = {wj : 1 ≤ j ≤ m} be basis for V and W -respectively. Then γ = {vi⊗ wj : 1 ≤ i ≤ n, 1 ≤ j ≤ m} forms a basis for V ⊗KW.
Hence dimK(V ⊗KW ) = mn.
Proof. Let us denote {fi : 1 ≤ i ≤ n} and {gj : 1 ≤ j ≤ n} the dual basis to α and β respectively.
Let us prove that γ is linearly independent. Assume that a =P
i,jaijvi⊗ wj = 0. Then a(fk, gl) =X
i,j
aij(vi⊗ wj)(fk, gl)
=X
i,j
aijfk(vi)gl(wj)
=X
i,j
aijδikδjl = akl
This shows that akl = 0 for 1 ≤ k ≤ n and 1 ≤ l ≤ m. In other words, γ is linearly independent.
Let c : V∗ × W∗ → K be a K-bilinear map. Since {fi} and {gj} are basis for V∗ and W∗ respectively, for every f ∈ V∗ and every g ∈ W∗, we can write f = P
iaifi and g =P
jbjgj. Then c(f, g) = P
i,jaibjc(fj, gj). Notice that vi(f ) = ai and wj(g) = bj. We see that (vi⊗ wj)(f, g) = aibj. This implies that
c(f, g) =
X
ij
c(fi, gj)vi⊗ wj
(f, g)
for all f ∈ V∗ and g ∈ W∗. This shows that c = P
i,jc(fi, gj)vi⊗ wj. In other words, all vectors in V ⊗KW is a linear span of γ. We conclude that γ forms a basis for V ⊗KW. We denote Tsr(V ) = ⊗ri=1V N ⊗sj=1V∗. Vectors of Tsr(V ) are called tensor of type (r, s).
Let {e1, · · · , en} be a basis for V and {θ1, · · · , θn} be its dual basis.
Lemma 1.2. The set {ei1⊗ · · · eir⊗ θj1⊗ · · · ⊗ θjs.} forms a basis for Tsr(V ).
Proof. The proof is similar to that given above. Any tensor T of type (r, s) can be repre- sented by
T =X
i,j
Tji1···ir
1···jsei1 ⊗ · · · eir ⊗ θj1 ⊗ · · · ⊗ θjs, where Tji11···j···isr = T (θi1, · · · , θir, ej1, · · · , ejs).
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