Algebra of endomorphisms

Description

Let $F$ be a field and let $V$ be a vector space over $F$. The algebra of endomorphisms $\End_F(V)$ is the set of endomorphisms of $V$, with pointwise addition, scalar multiplication, and composition as multiplication. It is an associative algebra over $F$, with the identity map $\id_V$ as unit. Thus $S(T + U) = ST + SU$, $(S + T)U = SU + TU$ and $a(ST) = (aS)T = S(aT)$ for all $S, T, U \in \End_F(V)$ and $a \in F$.

Remarks

If $\dim V = n$, then the choice of a basis gives an algebra isomorphism $\End_F(V) \cong M_n(F)$. For $\dim V \ge 2$, the algebra is not commutative. Its group of units is the group $GL(V)$ of linear isomorphisms $V \to V$.

Explanations

L1 human

The endomorphisms of a space can be added, scaled and composed. Composition plays the role of multiplication. This makes them a ring that is also a vector space, like the square matrices.