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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $T \\colon V \\to V$ be an [[endomorphism|0J7H0YBFDJ]].\nLet $p(x) = c_0 + c_1 x + \\cdots + c_d x^d$ be a polynomial in $F[x]$.\nThe *value of $p$ at $T$* is the endomorphism\n$$p(T) = c_0 \\id_V + c_1 T + c_2 T^2 + \\cdots + c_d T^d,$$\nwhere $T^k$ is the composition of $k$ copies of $T$.\nFor an $n \\times n$ matrix $A$, $p(A) = c_0 I_n + c_1 A + \\cdots + c_d A^d$ in the same way.\n\n## Remarks\n\nThe map $p \\mapsto p(T)$ is a homomorphism of algebras $F[x] \\to \\End_F(V)$: $(p + q)(T) = p(T) + q(T)$ and $(pq)(T) = p(T) q(T)$.\nIn particular, any two polynomials in $T$ commute.\nIf $T(v) = \\lambda v$, then $p(T)(v) = p(\\lambda) v$.\n\n## Explanations\n\n### L1 human\n\nWe can put a matrix into a polynomial, because matrices can be added, scaled and multiplied.\nThe constant term becomes a multiple of the identity matrix.\nFor example, if $p(x) = x^2 - 3x + 2$, then $p(A) = A^2 - 3A + 2I$.\n",
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