Tensor product of vector spaces

Definition

Let $F$ be a field, and let $V$ and $W$ be vector spaces over $F$. Let $L$ be the vector space over $F$ with basis the set $V \times W$; write $[v, w]$ for the basis vector of $(v, w)$. Let $N$ be the subspace of $L$ spanned by all elements of these forms, for $v, v' \in V$, $w, w' \in W$ and $a \in F$: $$[v + v', w] - [v, w] - [v', w], \qquad [v, w + w'] - [v, w] - [v, w'],$$ $$[av, w] - a[v, w], \qquad [v, aw] - a[v, w].$$ The tensor product of $V$ and $W$ is the quotient space $V \otimes W = L / N$. For $v \in V$ and $w \in W$, the class of $[v, w]$ is written $v \otimes w$.

Remarks

The map $(v, w) \mapsto v \otimes w$ is bilinear, and the vectors $v \otimes w$ span $V \otimes W$. Not every vector of $V \otimes W$ is of the form $v \otimes w$; for example, $e_1 \otimes e_1 + e_2 \otimes e_2$ in $F^2 \otimes F^2$ is not. The tensor product is characterised by its universal property. For finite-dimensional spaces, $V^* \otimes W$ is isomorphic to the space of linear maps from $V$ to $W$.

Explanations

L1 human

The tensor product combines two vector spaces into one space of formal products $v \otimes w$. These products follow the rules of bilinearity and nothing more. It turns bilinear maps into linear maps, which are easier to study.