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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 *characteristic polynomial* of $T$ is\n$$p_T(t) = p_{[T]_B}(t) = \\det(t I_n - [T]_B),$$\nthe [[characteristic polynomial|0VSSPAFM2C]] of the [[matrix of $T$|0YQSGNYE3K]] in a basis $B$ of $V$, where $n = \\dim V$.\n\n## Remarks\n\nThe polynomial does not depend on $B$, see [[the invariance of the characteristic polynomial|0TYMASG33Y]].\nIts degree is $\\dim V$.\nFor the identity map of $V$, it is $(t - 1)^n$.\n\n## Explanations\n\n### L1 human\n\nA linear map has a matrix in each basis.\nAll these matrices have the same characteristic polynomial.\nSo we can call it the characteristic polynomial of the map.\n",
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