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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \\colon V \\to V$ be an [[endomorphism|0J7H0YBFDJ]].\nThe *determinant* of $T$ is\n$$\\det T = \\det [T]_B,$$\nthe [[determinant|00KDZRW4C0]] of the [[matrix of $T$|0YQSGNYE3K]] in a basis $B$ of $V$.\n\n## Remarks\n\nThe value does not depend on $B$, because another basis gives a [[similar|0E89HQATM8]] matrix $P^{-1}[T]_B P$, with the same determinant.\nIt satisfies $\\det(S \\circ T) = \\det S \\cdot \\det T$ and $\\det \\id_V = 1$.\nThe map $T$ is invertible if and only if $\\det T \\ne 0$.\nFor $V = 0$, the convention is $\\det T = 1$.\n\n## Explanations\n\n### L1 human\n\nWe define the determinant of a linear map with its matrix in some basis.\nAnother basis changes the matrix, but not the determinant.\nSo the determinant belongs to the map itself, like the trace.\n",
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