Editing Geometric algebra (section)

=== Representation of subspaces ===
{{see also|Grassmannian}}
Geometric algebra represents subspaces of <math>V</math> as blades, and so they coexist in the same algebra with vectors from {{tmath|1= V }}. A {{tmath|1= k }}-dimensional subspace <math>W</math> of <math>V</math> is represented by taking an orthogonal basis <math>\{b_1,b_2,\ldots, b_k\}</math> and using the geometric product to form the [[blade (geometry)|blade]] {{tmath|1= D = b_1b_2\cdots b_k }}. There are multiple blades representing {{tmath|1= W }}; all those representing <math>W</math> are scalar multiples of {{tmath|1= D }}. These blades can be separated into two sets: positive multiples of <math>D</math> and negative multiples of {{tmath|1= D }}. The positive multiples of <math>D</math> are said to have ''the same [[orientation (vector space)|orientation]]'' as {{tmath|1= D }}, and the negative multiples the ''opposite orientation''.

Blades are important since geometric operations such as projections, rotations and reflections depend on the factorability via the exterior product that (the restricted class of) {{tmath|1= n }}-blades provide but that (the generalized class of) grade-{{tmath|1= n}} multivectors do not when {{tmath|1= n \ge 4 }}.