Difference between revisions of "Algebra (linear algebra)"

Revision as of 00:09, 2 April 2016

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 $F$) is a vector space, $(V,F)$, endowed with a bilinear map used for the product operation on the vector space^[1], that is a vector space $(V,F)$ with a map:

• $P:V\times V\rightarrow V$, which is bilinear^{[Recall 1]} called the "product".
• So now we define $xy:=P(x,y)$ 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:

Examples

• The class of smooth real-valued functions on $\mathbb{R}^n$

(See also the category: Examples of algebras)

TODO: Investigate the product of $k$-differentiable real valued functions

Recall notes

1. Jump up ↑ Recall that for a map to be bilinear we require:

   1. $P(\alpha x+\beta y,z)=\alpha P(x,z)+\beta P(y,z)$ and
   2. $P(x,\alpha y+\beta z)=\alpha P(x,y)+\beta P(x,z)$ for all $\alpha,\beta\in F$ and for all $x,y\in V$

References

1. ↑ ^{Jump up to: 1.0 1.1 1.2} Introduction to Smooth Manifolds - John M. Lee - Second Edition - Springer GTM