# A basis of identities of the algebra of third-order matrices by Genov G.K.

By Genov G.K.

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With b ∧ a. R x⊥ R−1 = = = = = = (b · a + b ∧ a) x⊥ (b · a + a ∧ b) ((b · a) x⊥ + b ∧ a ∧ x⊥ ) (b · a + a ∧ b) (x⊥ (b · a) + x⊥ ∧ b ∧ a) (b · a + a ∧ b) x⊥ (b · a + b ∧ a) (b · a + a ∧ b) x⊥ R R−1 x⊥ . Since x⊥ is not affected, we have determined that the rotation must be in the b ∧ a-plane. It remains to determine the angle of rotation in that plane. The rotation is composed of two reflections, which are orthogonal (angle preserving) transformations. So we can pick any vector in the b ∧ a-plane to determine that angle.

40) This defines a so-called Minkowski metric Rn+1,1 which is also used in physics for space-time. Note that the basis we use is not orthogonal. An alternative basis for the conformal model uses two vectors named e and e¯, with e · e = 1 and e¯ · e¯ = −1. This leads to an orthogonal metric matrix. The relation between these vectors and o and ∞ is 1 o = √ (e + e¯), 2 1 ∞ = √ (¯ e − e). 2 50 Since we are mainly concerned with Euclidean geometry, using the {e, e¯}-basis complicates matters because they have less geometrical significance in Euclidean geometry: e and e¯ represent spheres, while o and ∞ represent the origin and infinity, respectively3 .

A basis vector is either present or not present in such a combination. This leads to 2n elements (binary counting). 2, where we use it to form for the bitmap representation of basis blades. Since a multivector is a sum of blades, and blades can be written as a sum of basis blades, any multivector can be decomposed on a basis of blades. 9 Grade Part Selection It is useful to extract part of a multivector, based on grade. If A is the sum of homogeneous multivectors, as in A = A0 + A1 + A2 + . . + An , then the notation A i means to select or extract the grade i part of A: A i = Ai .