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blade (Definition)

A blade is a term often used to describe a basis entity in the space defined by a geometric algebra. Since a geometric algebra is a multi-graded space, the basis entities also have multiple grades. To distinguish the various graded entities, the blades are often prefixed by their grade. For example a grade-$ k$ basis entity would be called a $ k$-blade.

The number of linearly independent $ k$-blades in a particular geometric algebra is dependent on the number of dimensions of the manifold on which the algebra is defined. For an $ n$-dimensional manifold, the number of $ k$-blades is given by the binomial coefficient.

$\displaystyle N_k = \left( \begin{array}{c} n \ k \end{array} \right) $
The total number of basis blades of all grades in a geometric algebra defined on an $ n$-manifold is then:
$\displaystyle N = \sum_{k=0}^n N_k = 2^n $



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See Also: basis, unit vector

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Cross-references: binomial coefficient, algebra, manifold, dimensions, linearly independent, number, multiple, geometric algebra, basis, term
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This is version 2 of blade, born on 2006-06-10, modified 2007-07-02.
Object id is 7994, canonical name is Blade.
Accessed 1415 times total.

Classification:
AMS MSC15A66 (Linear and multilinear algebra; matrix theory :: Clifford algebras, spinors)
 15A75 (Linear and multilinear algebra; matrix theory :: Exterior algebra, Grassmann algebras)
 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)

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