Algebra of endomorphisms of a vector space ↩

Metadata 
aliases: [Algebra of vector space endomorphisms]
shorthands: {"End": "{\text{End}(A)}"}
created: 2022-01-09 15:48:20
modified: 2022-01-16 13:48:06

, the set of all endomorphisms of a vector space form an algebra.

Notation

is the set of all endomorphisms of a vector space.

Making into an algebra

Vector space properties

In order to form an algebra, needs to be a vector space over field first. Let .
Then is defined like:

For and we define by:

The operation is easy to see that is associative and commutative.
The zero element of is the endomorphism sending all to the zero element in .
The additive inverse of is defined by:

Clearly is the zero endomorphism.
The other properties are easy to verify so is a vector space over .

Algebra properties

We want to make the vector space into an algebra, so we need to have an additional bilinear multiplication as well. The product can be just defined like this:

So it is just the composition of the endomorphisms. The linearity of is easy to verify.

The multiplication also distributes over addition. For and :

Thus is an algebra.

The unit element

This algebra has a unit element: the identity map . It is denoted by 1:

Properties of the units of