---
tag: 03F5KP94K0
kind: definition
name: Determinant of a linear map
msc: [15A04, 15A15]
defines: 0X0HQYSZ0K
uses: [0C3CFHDN6R, 0PTF1PTJK0, 0YYMY95FWS]
sources:
  - url: https://en.wikipedia.org/wiki/Determinant
    locator: Determinant of an endomorphism
---

## Definition

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
The *determinant* of $T$ is
$$\det T = \det [T]_B,$$
the [[determinant|00KDZRW4C0]] of the [[matrix of $T$|0YQSGNYE3K]] in a basis $B$ of $V$.

## Remarks

The value does not depend on $B$, because another basis gives a [[similar|0E89HQATM8]] matrix $P^{-1}[T]_B P$, with the same determinant.
It satisfies $\det(S \circ T) = \det S \cdot \det T$ and $\det \id_V = 1$.
The map $T$ is invertible if and only if $\det T \ne 0$.
For $V = 0$, the convention is $\det T = 1$.

## Explanations

### L1 human

We define the determinant of a linear map with its matrix in some basis.
Another basis changes the matrix, but not the determinant.
So the determinant belongs to the map itself, like the trace.