---
tag: 0V94H674MB
kind: definition
name: Determinant of a family of vectors in a basis
msc: [15A15, 15A75]
defines: 05CJSSXPR3
uses: [0C3CFHDN6R, 0VTQTPQGAV, 0BMVBHTWV2]
sources:
  - url: https://en.wikipedia.org/wiki/Determinant
    locator: Determinant of n vectors
---

## Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $B = (b_1, \dots, b_n)$ be a [[basis|0KCQ9Y3QKA]] of $V$.
For vectors $v_1, \dots, v_n \in V$, let $M$ be the $n \times n$ matrix whose column $j$ is the [[coordinate vector|04CYQHN6MC]] $[v_j]_B$.
The *determinant of $(v_1, \dots, v_n)$ in the basis $B$* is
$$\det_B(v_1, \dots, v_n) = \det M.$$

## Remarks

The map $\det_B \colon V^n \to F$ is the unique alternating multilinear form with $\det_B(b_1, \dots, b_n) = 1$.
The vectors $v_1, \dots, v_n$ form a basis of $V$ if and only if $\det_B(v_1, \dots, v_n) \ne 0$.
For another basis $B'$, $\det_{B'} = \det_{B'}(b_1, \dots, b_n) \cdot \det_B$.

## Explanations

### L1 human

Write $n$ vectors in the coordinates of a fixed basis, and put the coordinates side by side as columns.
The determinant of this square matrix measures the signed volume of the box that the vectors span.
The unit is the box of the basis vectors.