Tensor Product and Linear Algebra

Tensor products can be rather intimidating for first-timers, so we’ll start with the simplest case: that of vector spaces over a field K. Suppose V and W are finite-dimensional vector spaces over K, with bases \{v_1, \ldots, v_n\} and \{w_1, \ldots, w_m\} respectively. Then the tensor product V\otimes_K W is the vector space with abstract basis \{ v_i w_j\}_{1\le i \le n, 1\le j\le m}. In particular, it is of dimension mn over K. Now we can “multiply” elements of V and W to obtain an element of this new space, e.g.

(2v_1 + 3v_2)(w_1 - 2w_3) = 2v_1 w_1 + 3 v_2 w_1 - 4 v_1 w_3 - 6v_2 w_3.

For example, if V is the space of polynomials in x of degree ≤ 2 and W is the space of polynomials in y of degree ≤ 3, then V\otimes_K W is the space of polynomials spanned by x^i y^j where 0≤i≤2, 0≤j≤3. However, defining the tensor product with respect to a chosen basis is rather unwieldy: we’d like a definition which only depends on V and W, and not the bases we picked.

Definition. A bilinear map of vector spaces is a map B:V \times W \to X, where V, W, X are vector spaces, such that

  • when we fix w, B(-, w): V→X is linear;
  • when we fix v, B(v, -): W→X is linear.

The tensor product of V and W, denoted V\otimes_K W, is defined to be a vector space together with a bilinear map \psi : V\times W \to V\otimes_K W such that the following universal property holds:

  • for any bilinear map B: V\times W \to X, there is a unique linear map f:V\otimes_K W \to X such that f\circ \psi = B.

For v∈V and w∈W, the element v\otimes w := \psi(v,w) is called a pure tensor element.

The universal property guarantees that if the tensor product exists, then it is unique up to isomorphism. What remains is the

Proof of Existence.

Recall that if S is a basis of vector space V, then any linear function VW uniquely corresponds to a function SW. Thus if we let T be the (infinite-dimensional) vector space with basis:

\{e_{v, w} : v \in V, w\in W\}

then linear maps gTX correspond uniquely to functions BV×W → X. Saying that B is bilinear is precisely the same as g factoring through the subspace U to obtain \overline g : T/U \to X, where U is the subspace generated by elements of the form:

\begin{aligned} e_{v+v', w} - e_{v,w} - e_{v', w}, \qquad & e_{cv, w} - c\cdot e_{v,w}\\ e_{v, w+w'} - e_{v,w} - e_{v, w'},\qquad & e_{v,cw} - c\cdot e_{v,w}\end{aligned}

for all vv’ ∈ Vww’ ∈ W and constant c ∈ K. Hence T/U is precisely our desired vector space, with \psi : V\times W \to T/U given by (v, w) \mapsto e_{v,w} \pmod U. And vw is the image of e_{v,w} in T/U. ♦


From the proof, it is clear that V ⊗ W is spanned by the pure tensors; in general though, not every element of V ⊗ W is a pure tensor. E.g. vw + v’w’ is generally not a pure tensor. However, vw + vw’ + v’w + v’w’ = (v+v’)⊗(w+w’) is a pure tensor since ψ is bilinear.


Properties of Tensor Product

We have:

Proposition. The following hold for K-vector spaces:

  • K \otimes_K V \cong V, where c\otimes v\mapsto cv;
  • V \otimes_K W \cong W \otimes_K V, where v\otimes w\mapsto w\otimes v;
  • V \otimes_K (W \otimes_K W') \cong (V\otimes_K W)\otimes_K W', where v\otimes (w\otimes w') \mapsto (v\otimes w)\otimes w';
  • V \otimes_K (\oplus_i W_i) \cong \oplus_i (V\otimes W_i), where v\otimes (w_i)_i \mapsto (v\otimes w_i)_i.


For the first property, the map K × V → V taking (cv) to cv is bilinear over K, so by the universal property of tensor products, this induces fK ⊗ V → V taking cv to cv. On the other hand, let’s take the linear map gV → K ⊗ V mapping v to 1⊗v. It remains to prove gf and fg are identity maps. Indeed: fg takes v → 1⊗v → v and gf takes cv → cv → 1⊗cvcv where the equality follows from bilinearity of ⊗.

For the third property, fix vV. The map W×W’ → (VW)⊗W’ taking (ww’) to (vw)⊗w‘ is bilinear in W and W’ so it induces f_v : W\otimes W' \to (V\otimes W)\otimes W' taking w\otimes w' \mapsto (v\otimes w)\otimes w'. Next we check that the map

V\times (W\otimes W') \to (V\otimes W)\otimes W', \qquad (v, x) \mapsto f_v(x)

is bilinear so it induces a linear map f : V\otimes (W\otimes W') \mapsto (V\otimes W)\otimes W' taking v\otimes (w\otimes w') \mapsto (v\otimes w)\otimes w'. Similarly one defines a reverse map g: (V\otimes W)\otimes W' \to V\otimes (W\otimes W') taking (v\otimes w)\otimes w' \mapsto v\otimes (w\otimes w'). Since the pure tensors generate the whole space, it follows that f and g are mutually inverse.

The second and fourth properties are left to the reader. ♦

As a result of the second and fourth properties, we also have:

Corollary. For any collection \{V_i\} and \{W_j\} of vector spaces, we have:

\oplus_{i, j} (V_i \otimes_K W_j) \cong (\oplus_i V_i)\otimes_K (\otimes_j W_j),

where the LHS element (v_i) \otimes (w_j) maps to (v_i \otimes w_j)_{i,j} on the RHS.

In particular, if \{v_i\} and \{w_j\} are bases of V and W respectively, then

V = \oplus_i Kv_i, \ W = \oplus_j Kw_j \implies V\otimes W = \oplus_{i, j} K(v_i \otimes w_j)

so \{v_i \otimes w_j\} forms a basis of VW. This recovers our original intuitive definition of the tensor product!


Tensor Product and Duals

Recall that the dual of a vector space V is the space V* of all linear maps VK. It is easy to see that V* ⊕ W* is naturally isomorphic to (V ⊕ W)* and when V is finite-dimensional, V** is naturally isomorphic to V.

[ One way to visualize V** ≅ V is to imagine the bilinear map V* × V → K taking (fv) to f(v). Fixing f we obtain a linear map VK as expected while fixing v we obtain a linear map V*→K and this corresponds to an element of V**. ]

If V is finite-dimensional, then a basis \{v_1, \ldots, v_n\} of V gives rise to a dual basis \{f_1, \ldots, f_n\} of V* where

f_i(v_j) = \begin{cases} 1, \quad &\text{if } i = j,\\ 0,\quad &\text{otherwise.}\end{cases}

or simply f_i(v_j) = \delta_{ij} with the Kronecker delta symbol. The next result we would like to show is:

Proposition. Let V and W be finite-dimensional over K.

  • We have V^*\otimes W^* \cong (V\otimes W)^* taking (f, g) to the map V\otimes W\to K, (v\otimes w) \mapsto f(v)g(w).
  • Also V^* \otimes W \cong \text{Hom}_K(V, W) taking (f, w) to the map V\to W, v\mapsto f(v)w.


For the first case, fix fV*, gW*. The map V\times W \to K taking (v,w)\mapsto f(v)g(w) is bilinear so it induces a map h:V\otimes W\to K taking (v\otimes w)\mapsto f(v)g(w). But the assignment (fg) → h gives rise to a map V^* \times W^* \to (V\otimes W)^* which is bilinear so it induces \varphi:V^* \otimes W^* \to (V\otimes W)^*. Note that f\otimes g corresponds to the map h:V\otimes W\to K taking v\otimes w \mapsto f(v)g(w).

To show that this is an isomorphism, let \{v_i\} and \{w_j\} be bases of V and W respectively, with dual bases \{f_i\} and \{g_j\} of V* and W*. The map then takes f_i \otimes g_j to the linear map V\otimes W\to K which takes v_k \otimes w_l to f_i(v_k) g_j(w_l) = \delta_{ik}\delta_{jl}. But this corresponds to the dual basis of \{v_i \otimes w_j\}, so we see that the above map φ takes a basis \{f_i \otimes g_j\} to a basis: dual of \{v_i\otimes w_j\}.

The second case is left as an exercise. ♦


Here’s one convenient way to visualize the above. Suppose elements of V comprise of column vectors. Then V* is the space of row vectors, and evaluating V* × → K corresponds to multiplying a row vector by column vector, thus giving a scalar. So V*W* ≅ (VW)* follows quite easily: indeed, the LHS concatenates two spaces of row vectors, while the RHS concatenates two spaces of column vectors then turns it into a space of row vectors.

The tensor product is a little trickier: for V and W we take column vectors with entries \alpha_1, \ldots, \alpha_n and \beta_1, \ldots, \beta_m respectively. Then we form the column vector with mn entries \alpha_i \beta_j. This lets us see why V*⊗W* ≅ (VW)*: in both cases we get a row vector with mn entries. Finally, to obtain V* ⊗W we take row vectors \alpha_1, \ldots, \alpha_n for elements of V* and column vectors \beta_1, \ldots, \beta_m for those of W, and the these multiply to give us an m × n matrix, which represents linear maps VW:



Consider the map V* × V → K which takes (f, v) to f(v). This is bilinear so it induces a linear map fV*⊗V → K. On the other hand, V*⊗V is naturally isomorphic to End(V), the space of K-linear maps VV. If we represent elements of End(V) as square matrices, what does f correspond to?

[ Answer: the trace of the matrix. ]

Tensor Algebra

Given a vector space V, let us consider n consecutive tensors:

V^{\otimes n} := \overbrace{V\otimes V\otimes \ldots \otimes V}^{n \text{ copies}}.

and let T(V) be the direct sum \oplus_{n=0}^\infty V^{\otimes n} = K \oplus V \oplus (V\otimes V) \oplus (V\otimes V\otimes V)\ldots. This gives an associative algebra over K by extending the bilinear map

V^{\otimes m} \times V^{\otimes n} \to V^{\otimes (m+n)}, \quad (v_1, v_2) \mapsto v_1 \otimes v_2.

to the entire space T(V) × T(V) → T(V). Note that it is not commutative in general. For example, suppose V has a basis {xyz}. Then

  • V^{(2)} has basis \{x^2, xy, xz, yx, y^2, yz, zx, zy, z^2\}, where we have shortened the notation x^2 := x\otimes x, xy := x\otimes y, etc.
  • V^{(3)} has basis \{x^3, x^2 y, \ldots\}, with 27 elements.
  • Multiplying V\times V^{(2)} \to V^{(3)} gives (x+z)(xy + zx) = x^2 y + xzx + zxy + z^2 x.

The algebra T(V), called the tensor algebra of V, satisfies the following universal property.

Theorem. The natural map ψ : V → T(V) is a linear map such that:

  • for any associative K-algebra A, and K-linear map φ: V → A, there is a unique K-algebra homomorphism f: T(V) → A such that φ = fψ.

Thus, \text{Hom}_{K-\text{lin}}(V, A) \cong \text{Hom}_{K-\text{alg}}(T(V), A).

However, often we would like multiplication to be commutative (e.g. when dealing with polynomials) and we’ll use the symmetric tensor algebra instead. Or we would like multiplication to be anti-commutative, i.e. xy = –yx (e.g. when dealing with differential forms) and we’ll use the exterior tensor algebra instead. We will say more about these when the need arises.


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