---
tag: 0MM085RX00
kind: definition
name: Product of vector spaces
msc: [15A03]
defines: 0EX2HPXPJ6
uses: [05EGHEQZAW]
sources:
  - url: https://en.wikipedia.org/wiki/Direct_sum_of_modules
    locator: Construction for two vector spaces
---

## Definition

Let $F$ be a field and let $V$ and $W$ be [[vector spaces|000AYPA9AJ]] 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|0S7W6EG32V]], written $V \oplus W$.
The [[dimension of a direct product|06HN47VPTH]] 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 $\R$.