Let $V$ be a [vector space](/page/Vector%20Space) over a field $F$, and let $\operatorname{End}_F(V)$ denote the set of $F$-linear maps $V \to V$. Define addition, scalar multiplication, and multiplication on $\operatorname{End}_F(V)$ as follows: for $S,T \in \operatorname{End}_F(V)$, $a \in F$, and $v \in V$,
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\begin{align*}
(S+T)(v)=S(v)+T(v).
\end{align*}
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\begin{align*}
(aT)(v)=aT(v).
\end{align*}
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\begin{align*}
(ST)(v)=S(T(v)).
\end{align*}
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Then $\operatorname{End}_F(V)$ is a unital associative $F$-algebra under these operations, with multiplicative identity $\operatorname{id}_V$.