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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 [[trace of $T$|0KZDNGA561]] equals the trace of the [[matrix of $T$|0YQSGNYE3K]] in $B$:\n$$\\tr T = \\tr [T]_B.$$\n\n## Remarks\n\nThe key fact is that similar matrices have the same trace, by $\\tr(AB) = \\tr(BA)$.\nWith a basis-free definition, the trace is the map $\\End_F(V) \\cong V^* \\otimes V \\to F$ induced by evaluation $f \\otimes v \\mapsto f(v)$.\n\n## Explanations\n\n### L1 human\n\nWe can compute the trace of a linear map in any basis that we like.\nWe write the matrix in that basis and add its diagonal entries.\nEvery basis gives the same number.\n",
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