---
tag: 0QW084460E
kind: statement
role: proposition
name: Dimension of a subspace
msc: [15A03]
uses: [05G2G0P224, 0EPTJA5QV4]
sources:
  - url: https://en.wikipedia.org/wiki/Dimension_(vector_space)
    locator: Properties
---

## Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $W$ be a [[linear subspace|00YC1D652K]] of $V$.
Then $W$ is finite-dimensional, and $\dim W \le \dim V$, where $\dim$ is the [[dimension|03GSZ8RTZJ]].

## Remarks

A basis of $W$ is a linearly independent family in $V$, so it has at most $\dim V$ vectors.
Equality holds only when $W = V$, see [[a subspace of full dimension is the whole space|0WGAXEK99E]].
For infinite-dimensional $V$, the inequality $\dim W \le \dim V$ of cardinals still holds.

## Explanations

### L1 human

A subspace cannot have more independent directions than the space around it.
A plane in ordinary space has dimension $2$, which is at most $3$.
The only subspace with the same dimension is the whole space.