Product of vector spaces

Definition

Let $F$ be a field and let $V$ and $W$ be vector spaces over $F$. The product $V \times W$ is the set of pairs $(v, w)$ with $v \in V$ and $w \in W$, with the operations $$(v, w) + (v', w') = (v + v', w + w'), \qquad a(v, w) = (av, aw)$$ for $v, v' \in V$, $w, w' \in W$ and $a \in F$. It is a vector space over $F$ with zero vector $(0, 0)$.

Remarks

For finitely many spaces, the product is also the external direct sum, written $V \oplus W$. The dimension of a direct product is $\dim V + \dim W$. For example, $F^m \times F^n$ is isomorphic to $F^{m+n}$. The projections $(v, w) \mapsto v$ and $(v, w) \mapsto w$ are linear maps.

Explanations

L1 human

The product of two vector spaces has vectors that are pairs: one vector from each space. We add pairs part by part and scale both parts. The plane $\R^2$ is the product of two copies of the line $\R$.