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  "body": "\n## Statement\n\nLet $R$ be a commutative ring, let $A$ and $B$ be $n \\times n$ matrices over $R$, and let $c \\in R$.\nThen $\\tr(A + B) = \\tr A + \\tr B$ and $\\tr(cA) = c \\tr A$, where $\\tr$ is the [[trace|0XC1RDBY7A]].\n\n## Remarks\n\nThus the trace is a linear map from $M_n(R)$ to $R$.\nBoth identities follow at once from the definition, because the diagonal of $A + B$ is the sum of the diagonals.\nOver a field, the trace is a [[linear form|07YQ1C78G0]] on the vector space of $n \\times n$ matrices.\n\n## Explanations\n\n### L1 human\n\nThe trace of a sum is the sum of the traces.\nThe trace of a multiple is the same multiple of the trace.\nThis holds because we only add diagonal entries, and addition and scaling act entry by entry.\n",
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