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  "body": "\n## Summary\n\nLet $n \\ge 1$ and let $v_1, \\dots, v_n \\in \\R^n$ be the columns of a real $n \\times n$ matrix $A$.\nThe parallelepiped spanned by them is $P = \\{t_1 v_1 + \\cdots + t_n v_n : 0 \\le t_i \\le 1\\}$.\nIts $n$-dimensional volume is $\\abs{\\det A}$, where $\\det$ is the [[determinant|00KDZRW4C0]].\nIf $\\det A \\ne 0$, then its sign is positive when $(v_1, \\dots, v_n)$ has the [[orientation|0QJJAQ6ZYZ]] of the standard basis, and negative when it has the opposite orientation.\n\n## Remarks\n\nIn $\\R^2$, $\\abs{ad - bc}$ is the area of the parallelogram spanned by $(a, c)$ and $(b, d)$.\nFor each region $S$ of finite volume, the linear map $x \\mapsto Ax$ multiplies the volume of $S$ by $\\abs{\\det A}$.\nThis fact is the origin of the Jacobian determinant in the change of variables formula.\n\n## Explanations\n\n### L1 human\n\nThe determinant measures how a matrix changes volume.\nThe unit cube goes to a box spanned by the columns, and $\\abs{\\det A}$ is the volume of this box.\nA negative determinant means that the matrix also turns space inside out, like a mirror.\n",
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