Determinant of a family of vectors in a basis

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 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 $[v_j]_B$. 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.