Linear and multilinear algebra; matrix theory Object

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 $V \to V$ .

Tag	`0E3X2G1ZQB`
URI	`m2://0E3X2G1ZQB`
Kind	object
Writing	W0 stub
MSC	15A04, 16S50
Instance of	Space of linear maps

Algebra of endomorphisms

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