Determinant

It has two different slightly different meanings:

The determinant of a matrix, which is the usual quantity from the high school.
The determinant of a linear endomorphism $T$ of a vector space $V$ .
It can be defined like the determinant of the matrix of the linear map with respect to any basis, in the sense of the definition above, since it can be shown that this quantity does not depends on the choice of the basis.
Or, we could give an intrinsic definition, by choosing any volume form $Ω$ and any collection of $n$ independent vectors $w_{i}$ . We then define

d e t (T) = \frac{Ω (T v_{1}, \dots, T v_{n})}{Ω (v_{1}, \dots, v_{n})}

Related: trace