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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]].\nA [[subspace|00YC1D652K]] $W$ of $V$ is *invariant* under $T$, or *$T$-invariant*, when $T(w) \\in W$ for all $w \\in W$.\n\n## Remarks\n\nThe subspaces $0$ and $V$, $\\ker T$ and $\\im T$, and each [[eigenspace|0G0RFPG8HR]] of $T$ are $T$-invariant.\nA line spanned by $v \\ne 0$ is $T$-invariant if and only if $v$ is an eigenvector of $T$.\nIf $V = W_1 \\oplus W_2$ with both $W_i$ invariant, then the matrix of $T$ in an adapted basis is block diagonal.\n\n## Explanations\n\n### L1 human\n\nAn invariant subspace is a part of the space that the map never leaves.\nVectors inside it stay inside it.\nWe can then study the map on this smaller space alone.\n",
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