https://www.maths.kisogo.com/index.php?title=Algebra_(linear_algebra)&feed=atom&action=historyAlgebra (linear algebra) - Revision history2024-03-29T14:25:18ZRevision history for this page on the wikiMediaWiki 1.24.1https://www.maths.kisogo.com/index.php?title=Algebra_(linear_algebra)&diff=3910&oldid=prevAlec: Cleaning it up!2017-01-01T05:26:09Z<p>Cleaning it up!</p>
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</table>Alechttps://www.maths.kisogo.com/index.php?title=Algebra_(linear_algebra)&diff=2148&oldid=prevAlec: Moving page from Algebra, last modified 21st Oct 2015, viewcount 862016-03-27T15:36:26Z<p>Moving page from <a href="/index.php?title=Algebra" title="Algebra">Algebra</a>, last modified 21st Oct 2015, viewcount 86</p>
<p><b>New page</b></p><div>: '''Note: ''' Not to be confused with [[Algebra of sets|an algebra of sets]] (as would be encountered in measure theory) see [[Algebra (Disambiguation)]] for all uses<br />
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==Definition==<br />
An ''algebra'' (over a [[field]] {{M|F}}) is a [[vector space]], {{M|(V,F)}}, endowed with a [[bilinear map]] used for the ''product'' operation on the vector space<ref name="ITSM">Introduction to Smooth Manifolds - John M. Lee - Second Edition - Springer GTM</ref>, that is a vector space {{M|(V,F)}} with a map:<br />
* {{M|1=P:V\times V\rightarrow V}}, which is [[bilinear map|bilinear]]<ref group="Recall">Recall that for a map to be bilinear we require:<br />
:# {{M|1=P(\alpha x+\beta y,z)=\alpha P(x,z)+\beta P(y,z)}} and<br />
:# {{M|1=P(x,\alpha y+\beta z)=\alpha P(x,y)+\beta P(x,z)}} for all {{M|\alpha,\beta\in F}} and for all {{M|x,y\in V}}</ref> called the "product". <br />
* So now we define {{M|1=xy:=P(x,y)}} on the space, thus endowing our vector space with a notion of product.<br />
===Properties===<br />
We may say an ''algebra'' is any ''(zero or more)'' of the following if it satisfies the definitions:<br />
{| class="wikitable" border="1"<br />
|-<br />
! Property<br />
! Definition<br />
|-<br />
! Commutative<ref name="ITSM"/><br />
| If the product is commutative. That is if {{M|1=xy=yx}} (which is the same as {{M|1=P(x,y)=P(y,x)}})<br />
|-<br />
! Associative<ref name="ITSM"/><br />
| If the product is associative. That is if {{M|1=x(yz)=(xy)z}} (which is the same as {{M|1=P(x,P(y,z))=P(P(x,y),z)}})<br />
|}<br />
==Examples==<br />
* [[Class of smooth real-valued functions on R-n|The class of smooth real-valued functions on {{M|\mathbb{R}^n}}]]<br />
(See also the category: [[:Category:Examples of algebras|Examples of algebras]])<br />
{{Todo|{{Highlight|Investigate the product of {{M|k}}-differentiable real valued functions}}}}<br />
==Recall notes==<br />
<references group="Recall"/><br />
==References==<br />
<references/><br />
{{Definition|Manifolds|Differential Geometry|Functional Analysis}}</div>Alec