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  "body": "\n## Description\n\nLet $F$ be a field and let $V$ be a vector space over $F$.\nThe *algebra of endomorphisms* $\\End_F(V)$ is the set of [[endomorphisms|0J7H0YBFDJ]] of $V$, with pointwise addition, scalar multiplication, and composition as multiplication.\nIt is an associative algebra over $F$, with the identity map $\\id_V$ as unit.\nThus $S(T + U) = ST + SU$, $(S + T)U = SU + TU$ and $a(ST) = (aS)T = S(aT)$ for all $S, T, U \\in \\End_F(V)$ and $a \\in F$.\n\n## Remarks\n\nIf $\\dim V = n$, then the choice of a basis gives an algebra isomorphism $\\End_F(V) \\cong M_n(F)$.\nFor $\\dim V \\ge 2$, the algebra is not commutative.\nIts group of units is the group $GL(V)$ of linear isomorphisms $V \\to V$.\n\n## Explanations\n\n### L1 human\n\nThe endomorphisms of a space can be added, scaled and composed.\nComposition plays the role of multiplication.\nThis makes them a ring that is also a vector space, like the square matrices.\n",
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