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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $B = (b_1, \\dots, b_n)$ be a [[basis|0KCQ9Y3QKA]] of $V$.\nFor vectors $v_1, \\dots, v_n \\in V$, let $M$ be the $n \\times n$ matrix whose column $j$ is the [[coordinate vector|04CYQHN6MC]] $[v_j]_B$.\nThe *determinant of $(v_1, \\dots, v_n)$ in the basis $B$* is\n$$\\det_B(v_1, \\dots, v_n) = \\det M.$$\n\n## Remarks\n\nThe map $\\det_B \\colon V^n \\to F$ is the unique alternating multilinear form with $\\det_B(b_1, \\dots, b_n) = 1$.\nThe vectors $v_1, \\dots, v_n$ form a basis of $V$ if and only if $\\det_B(v_1, \\dots, v_n) \\ne 0$.\nFor another basis $B'$, $\\det_{B'} = \\det_{B'}(b_1, \\dots, b_n) \\cdot \\det_B$.\n\n## Explanations\n\n### L1 human\n\nWrite $n$ vectors in the coordinates of a fixed basis, and put the coordinates side by side as columns.\nThe determinant of this square matrix measures the signed volume of the box that the vectors span.\nThe unit is the box of the basis vectors.\n",
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