Tensor Product - Tensor Product of Vector Spaces

Tensor Product of Vector Spaces

The tensor product V ⊗K W of two vector spaces V and W over a field K can be defined by the method of generators and relations. (The tensor product is often denoted V ⊗ W when the underlying field K is understood.)

To construct V ⊗ W, one begins with the set of ordered pairs in the Cartesian product V × W. For the purposes of this construction, regard this Cartesian product as a set rather than a vector space. The free vector space F on V × W is defined by taking the vector space in which the elements of V × W are a basis. In set-builder notation,

where we have used the symbol e(v,w) to emphasize that these are taken to be linearly independent by definition for distinct (v, w) ∈ V × W.

The tensor product arises by defining the following four equivalence relations in F(V × W):

\begin{align} e_{(v_1 + v_2, w)} &\sim e_{(v_1, w)} + e_{(v_2, w)}\\ e_{(v, w_1 + w_2)} &\sim e_{(v, w_1)} + e_{(v, w_2)}\\ ce_{(v, w)} &\sim e_{(cv, w)} \sim e_{(v, cw)} \end{align}

where v, v1 and v2 are vectors from V, while w, w1, and w2 are vectors from W, and c is from the underlying field K. Denoting by R the space generated by these four equivalence relations, the tensor product of the two vector spaces V and W is then the quotient space

It is also called the tensor product space of V and W and is a vector space (which can be verified by directly checking the vector space axioms). The tensor product of two elements v and w is the equivalence class (e(v,w) + R) of e(v,w) in V ⊗ W, denoted v ⊗ w. This notation can somewhat obscure the fact that tensors are always cosets: manipulations performed via the representatives (v,w) must always be checked that they do not depend on the particular choice of representative.

The space R is mapped to zero in V ⊗ W, so that the above three equivalence relations become equalities in the tensor product space:

\begin{align} (v_1 + v_2) \otimes w &= v_1 \otimes w + v_2 \otimes w;\\ v \otimes (w_1 + w_2) &= v \otimes w_1 + v \otimes w_2;\\ cv \otimes w &= v \otimes cw = c(v \otimes w). \end{align}

Given bases {vi} and {wi} for V and W respectively, the tensors {vi ⊗ wj} form a basis for V ⊗ W. The dimension of the tensor product therefore is the product of dimensions of the original spaces; for instance Rm ⊗ Rn will have dimension mn.

Elements of V ⊗ W are sometimes referred to as tensors, although this term refers to many other related concepts as well. An element of V ⊗ W of the form v ⊗ w is called a pure or simple tensor. In general, an element of the tensor product space is not a pure tensor, but rather a finite linear combination of pure tensors. That is to say, if v1 and v2 are linearly independent, and w1 and w2 are also linearly independent, then v1 ⊗ w1 + v2 ⊗ w2 cannot be written as a pure tensor. The number of simple tensors required to express an element of a tensor product is called the tensor rank, (not to be confused with tensor order, which is the number of spaces one has taken the product of, in this case 2; in notation, the number of indices) and for linear operators or matrices, thought of as (1,1) tensors (elements of the space V ⊗ V*), it agrees with matrix rank.

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