Geometric Linear Algebra

In ??? Herman Grassmann invented Geometric Algebra…

Let E be the set of points in space.

Define \operatorname{Alg}(E) by

\boldsymbol{R}\subseteq \operatorname{Alg}(E)

E\subseteq \operatorname{Alg}(E)

If A,B\in\operatorname{Alg}(E) then A + B = B + A\in\operatorname{Alg}(E)

If A,B\in\operatorname{Alg}(E) then AB \in\operatorname{Alg}(E)

If a\in\boldsymbol{R} and A\in\operatorname{Alg}(E) then aA = Aa.

If A\in\operatorname{Alg}(E) then 0A = 0 and 1A = A.

If A,B,C\in\operatorname{Alg}(E) then A(B + C) = AB + AC and (A + B)C = AC + BC.

Graded.

Linear Transformation: T\colon E\to E. T(aP) = aTP, T(P + Q) = TP + TQ.

Grassmann: PQ = 0 if and only if P = Q.

P(\sum_i a_i P_i) = 0 if and only if P = \sum_i a_i P_i.

Independence: \sum_i a_i P_i = 0 implies a_i = 0 for all i.

P_0\cdots P_n = 0 iff (P_j) independent.

\partial P_0 P_1 P_2 = P_1 P_2 - P_0 P_2 + P_0 P_1 = (P_1 - P_0)(P_2 - P_0).

(P_1 - P_0)(P_2 - P_0) = 0 iff P_1 - P_0 = a(P_2 - P_0.