Geometric algebra: Difference between revisions

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'''Geometric algebra''' is ~~an other~~ name for [[wp:Clifford algebra|Clifford algebra]]s and it's basically an algebra containing a vector space <math>\mathcal{V}</math> and obeying the following axioms:

'''Geometric algebra''' is another name for [[wp:Clifford algebra|Clifford algebra]]s and it's basically an algebra containing a vector space <math>\mathcal{V}</math> and obeying the following axioms:

:<math>\begin{array}{c}

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The product operation in such algebra is called the ''geometric product''. Elements are called ''multivectors'', while multivectors in <math>\mathcal{V}</math> are just called ''vectors''.

There are a few simple examples of geometric algebras. A trivial one for instance is simply <math>\R</math>, where <math>\mathcal{V} = \R</math>. The complex numbers also form a geometric algebra, where the vector space is the one-dimensional space of all purely imaginary numbers. ~~An other~~ example is the space of [[Quaternion type|quaternions]], where the vector space is the three-dimensional space of all linear combinations of <math>(i, j, k)</math>.

There are a few simple examples of geometric algebras. A trivial one for instance is simply <math>\R</math>, where <math>\mathcal{V} = \R</math>. The complex numbers also form a geometric algebra, where the vector space is the one-dimensional space of all purely imaginary numbers. Another example is the space of [[Quaternion type|quaternions]], where the vector space is the three-dimensional space of all linear combinations of <math>(i, j, k)</math>.

The purpose of this task is to implement a geometric algebra with a vector space <math>\mathcal{V}</math> of dimension ''n'' of at least five, but for extra-credit you can implement a version with ''n'' arbitrary large. Using a dimension five is useful as it is the dimension required for the so-called ''conformal model'' which will be the subject of a derived task.