By Roger A. Johnson

This vintage textual content explores the geometry of the triangle and the circle, focusing on extensions of Euclidean thought, and reading intimately many rather contemporary theorems. numerous hundred theorems and corollaries are formulated and proved thoroughly; quite a few others stay unproved, for use via scholars as workouts. 1929 version.

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I can demonstrate directly that PAr (a) lies in Ar : −1 PAr (a) ∧ Ar = (a ⌋ Ar A−1 r ) ∧ Ar = a ⌋ Ar Ar Ar r+1 = a ⌋ Ar r+1 = 0. (225) r−1 = a ∧ Ar r−1 = 0. (226) Similarly, I can show that RAr (a) is orthogonal to Ar as follows: RAr (a) ⌋ Ar = (a ∧ Ar Ar−1 ) ⌋ Ar = a ∧ Ar Ar−1 Ar This result applies to the Gram-Schmidt process for producing an orthogonal set of vectors from a linearly independent set with the same span. ,r as follows. Let b1 = a1 to start with. Then b2 equals a2 minus its projection onto b1 , or equivalently the orthogonal rejection of a2 from b1 .

The commutator The final operation is called the commutator, defined as follows. A × B := 1 (AB − BA). 2 (188) Notice the factor of 21 , which is not present in the usual definition of the commutator, used for example in quantum mechanics. The commutator obeys the identity A × (BC) = (A × B)C + B(A × C), (189) which is easily verified by expanding out the commutators. This shows that the commutator is a derivation on the algebra (it obeys the Leibnitz rule). Use this identity to expand A × (BC) and A × (CB) and take half the difference; the result is the Jacobi identity A × (B × C) = (A × B) × C + B × (A × C).

The result is a × A = a⌋ A A×a= A + + a∧ A +⌊a+ A − − ∧ a. (193) This lets me prove an important result about commuting multivectors. Theorem 28. The following statements are equivalent. 1. A commutes with all multivectors. 2. A commutes with all vectors. 3. A = λ + µ I −. Item 3 is my sneaky way of saying A equals λ in even-dimensional vector spaces and λ + µI in odddimensional spaces. Proof. Since scalars commute with everything, I won’t mention them again. If A commutes with all multivectors then it obviously commutes with all vectors.