bladeBlade2006-06-10 10:20:572007-07-02 09:58:56Definition% this is the default PlanetMath preamble. as your knowledge
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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.
\[
N_k = \left( \begin{array}{c}
n \\
k
\end{array} \right)
\]
The total number of basis blades of all grades in a geometric algebra defined on an $n$-manifold is then:
\[
N = \sum_{k=0}^n N_k = 2^n
\]