# Algebra (linear algebra)

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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[1], 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[1] 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[1] 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

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]

## References

1. Introduction to Smooth Manifolds - John M. Lee - Second Edition - Springer GTM