Determinant of a linear map

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. The determinant of $T$ is $$\det T = \det [T]_B,$$ the determinant of the matrix of $T$ in a basis $B$ of $V$.

Remarks

The value does not depend on $B$, because another basis gives a similar 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.