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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 *trace* of $T$ is\n$$\\tr T = \\tr [T]_B,$$\nthe [[trace|0XC1RDBY7A]] of the matrix of $T$ in a basis $B$ of $V$.\n\n## Remarks\n\nThe value does not depend on $B$: by the [[change of basis formula|0MQXVZEBMT]], another basis gives $P^{-1} [T]_B P$, and $\\tr(P^{-1} A P) = \\tr A$.\nThe trace is a linear map $\\End_F(V) \\to F$, and $\\tr(ST) = \\tr(TS)$.\nThe trace of the identity map of $V$ is $\\dim V \\cdot 1$.\nA [[projection|033DA1WE8Y]] on a real or complex space has trace equal to its rank.\n\n## Explanations\n\n### L1 human\n\nWe define the trace of a linear map by its matrix in some basis.\nAnother basis gives a different matrix, but the trace stays the same.\nSo the trace belongs to the map, not to the basis.\n",
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