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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.\nA polynomial $p \\in F[x]$ *annihilates* $T$ when $p(T) = 0$, where $p(T)$ is the [[value of $p$ at $T$|0E3MV6E6MG]].\nSuch a polynomial is an *annihilating polynomial* of $T$.\n\n## Remarks\n\nThe annihilating polynomials of $T$ form an ideal of $F[x]$.\nIf $\\dim V = n$ is finite, then a nonzero annihilating polynomial exists, because the $n^2 + 1$ maps $\\id_V, T, \\dots, T^{n^2}$ are linearly dependent.\nThe monic generator of this ideal is the [[minimal polynomial|03NDVCSHSB]] of $T$.\nFor a [[projection|033DA1WE8Y]] $P$, the polynomial $x^2 - x$ annihilates $P$.\n\n## Explanations\n\n### L1 human\n\nAn annihilating polynomial is a polynomial that becomes the zero map when we put $T$ into it.\nFor example, a reflection $R$ satisfies $R^2 = I$, so $x^2 - 1$ annihilates it.\nSuch relations tell us a lot about the possible eigenvalues.\n",
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