---
tag: 08ZDRN6H41
kind: statement
role: lemma
name: Steinitz exchange lemma
aliases: [replacement-theorem]
msc: [15A03]
uses: [0EVHJFTG9Z, 0BJKVF90Z6]
sources:
  - url: https://en.wikipedia.org/wiki/Steinitz_exchange_lemma
    locator: Statement
same_as:
  wikidata: Q2338929
  wikipedia: Steinitz exchange lemma
---

## Statement

Let $F$ be a field and let $V$ be a vector space over $F$.
Let $u_1, \dots, u_m \in V$ be [[linearly independent|0CRMGPAH83]], and let $w_1, \dots, w_n \in V$ [[span|088RYKAWK9]] $V$.
Then $m \le n$.
Moreover, after a renumbering of the $w_j$, the family $u_1, \dots, u_m, w_{m+1}, \dots, w_n$ spans $V$.

## Remarks

The lemma follows by induction on $m$: each $u_i$ replaces one of the $w_j$ in a spanning family.
A consequence is that any two finite bases of $V$ have the same number of elements.
The lemma is named after Ernst Steinitz.

## Explanations

### L1 human

An independent family can never have more vectors than a spanning family.
We can trade the vectors of the spanning family, one by one, for the independent vectors, and the family still spans.
This lemma is the reason why the dimension of a space is well defined.