---
tag: 096V7H13H9
kind: statement
role: proposition
name: Linearly independent set of the right size is a basis
msc: [15A03]
uses: [0VTQTPQGAV, 05G2G0P224, 0EVHJFTG9Z]
sources:
  - url: https://en.wikipedia.org/wiki/Basis_(linear_algebra)
    locator: Properties
---

## Statement

Let $F$ be a field and let $V$ be a vector space over $F$ of finite [[dimension|03GSZ8RTZJ]] $n$.
Let $v_1, \dots, v_n \in V$ be $n$ vectors.
If $v_1, \dots, v_n$ are [[linearly independent|0CRMGPAH83]], then they form a [[basis|0KCQ9Y3QKA]] of $V$.
If $v_1, \dots, v_n$ span $V$, then they form a basis of $V$.

## Remarks

Thus, for exactly $\dim V$ vectors, independence and spanning are equivalent.
For example, three vectors of $\R^3$ form a basis if and only if the $3 \times 3$ matrix with these columns has nonzero determinant.
More generally, a family $v_1, \dots, v_k$ is linearly independent if and only if its span has dimension $k$.

## Explanations

### L1 human

When we have exactly as many vectors as the dimension, we only need to check one of the two basis conditions.
Independence gives spanning for free, and spanning gives independence for free.
This halves the work of checking a basis.