---
tag: 06HN47VPTH
kind: statement
role: proposition
name: Dimension of a direct product
msc: [15A03]
uses: [0MM085RX00, 05G2G0P224]
sources:
  - url: https://en.wikipedia.org/wiki/Direct_sum_of_modules
    locator: Properties
---

## Statement

Let $F$ be a field and let $V$ and $W$ be finite-dimensional vector spaces over $F$.
Then the [[product|0EX2HPXPJ6]] $V \times W$ is finite-dimensional, and
$$\dim(V \times W) = \dim V + \dim W,$$
where $\dim$ is the [[dimension|03GSZ8RTZJ]].

## Remarks

Let $(v_i)$ be a basis of $V$ and let $(w_j)$ be a basis of $W$.
Then the vectors $(v_i, 0)$ and $(0, w_j)$ together form a basis of $V \times W$.
For example, $\dim(\R^2 \times \R^3) = 5$.
By induction, $\dim(V_1 \times \cdots \times V_k) = \dim V_1 + \cdots + \dim V_k$.

## Explanations

### L1 human

A vector of $V \times W$ is a pair, so we need the directions of $V$ and the directions of $W$.
The dimensions add up.
The plane is the product of two lines, and $1 + 1 = 2$.