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  "body": "\n## Statement\n\nLet $F$ be a field, let $V$ be a finite-dimensional vector space over $F$ with a basis $B$, and let $T \\colon V \\to V$ be a linear map.\nThen the [[determinant of $T$|0X0HQYSZ0K]] equals the determinant of the [[matrix of $T$|0YQSGNYE3K]] in $B$:\n$$\\det T = \\det [T]_B.$$\n\n## Remarks\n\nThe key fact is $\\det(P^{-1} A P) = \\det A$, which follows from the product rule.\nOver a commutative ring, the same statement holds for free modules of finite rank.\n\n## Explanations\n\n### L1 human\n\nWe can compute the determinant of a linear map in any basis.\nWe write its matrix in that basis, and we take the determinant of the matrix.\nEvery basis gives the same answer.\n",
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