# Algebra (linear algebra)

Jump to: navigation, search
Grade: A
This page is currently being refactored (along with many others)
Please note that this does not mean the content is unreliable. It just means the page doesn't conform to the style of the site (usually due to age) or a better way of presenting the information has been discovered.
The message provided is:

• This page will be redone at Algebra

Note: Not to be confused with an algebra of sets (as would be encountered in measure theory) see Algebra (disambiguation) for all uses

## Definition

An algebra (over a field [ilmath]F[/ilmath]) is a vector space, [ilmath](V,F)[/ilmath], endowed with a bilinear map used for the product operation on the vector space, that is a vector space [ilmath](V,F)[/ilmath] with a map:

• [ilmath]P:V\times V\rightarrow V[/ilmath], which is bilinear[Recall 1] called the "product".
• So now we define [ilmath]xy:=P(x,y)[/ilmath] on the space, thus endowing our vector space with a notion of product.

### Properties

We may say an algebra is any (zero or more) of the following if it satisfies the definitions:

Property Definition
Commutative If the product is commutative. That is if [ilmath]xy=yx[/ilmath] (which is the same as [ilmath]P(x,y)=P(y,x)[/ilmath])
Associative If the product is associative. That is if [ilmath]x(yz)=(xy)z[/ilmath] (which is the same as [ilmath]P(x,P(y,z))=P(P(x,y),z)[/ilmath])

## Examples

(See also the category: Examples of algebras)

TODO: Investigate the product of [ilmath]k[/ilmath]-differentiable real valued functions

## Recall notes

1. Recall that for a map to be bilinear we require:
1. [ilmath]P(\alpha x+\beta y,z)=\alpha P(x,z)+\beta P(y,z)[/ilmath] and
2. [ilmath]P(x,\alpha y+\beta z)=\alpha P(x,y)+\beta P(x,z)[/ilmath] for all [ilmath]\alpha,\beta\in F[/ilmath] and for all [ilmath]x,y\in V[/ilmath]