endomorphism algebra
When the domain and codomain agree, the space of operators End(V) gains an extra operation you cannot perform on a general Hom(V,W): composition. Given two operators S and T, the composite S after T is again an operator. So End(V) is not merely a vector space — it is a vector space with a multiplication, which is exactly the structure called an algebra.
Spelled out, End(V) is closed under addition, scalar multiplication, and composition; composition distributes over addition on both sides; and the identity operator I acts as a multiplicative unit. These are precisely the axioms of an associative algebra with unit. The multiplication is associative because composing functions is associative, but it is NOT commutative: in general S after T differs from T after S.
Fix a basis of an n-dimensional V and the whole structure becomes concrete: End(V) is isomorphic to the algebra of n by n matrices, with composition matching matrix multiplication. Everything you know about matrix products — non-commutativity, the role of the identity matrix, invertibility — is the algebra structure of End(V) seen in coordinates.
This viewpoint pays off when you study a single operator T, because then the polynomials in T, namely things like a I + b T + c T^2, form a commutative subalgebra inside End(V). The minimal polynomial and the Cayley-Hamilton theorem are statements about that subalgebra, and they drive the canonical-form theory.
Picking a basis of an n-dimensional space identifies the endomorphism algebra with the algebra of n by n matrices over the field F.
End(V) has zero divisors when dim V > 1: two nonzero operators can compose to the zero operator (a projection onto a subspace times a projection onto a complementary one). So it is far from a field, and non-invertible elements abound.