Algebra (linear algebra)
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Grade: A
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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
(See also the category: Examples of algebras)
TODO: Investigate the product of [ilmath]k[/ilmath]-differentiable real valued functions
Recall notes
- ↑ Recall that for a map to be bilinear we require:
- [ilmath]P(\alpha x+\beta y,z)=\alpha P(x,z)+\beta P(y,z)[/ilmath] and
- [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
- ↑ ^{1.0} ^{1.1} ^{1.2} Introduction to Smooth Manifolds - John M. Lee - Second Edition - Springer GTM