Algebras {geometric algebra} can represent real and complex vector-space non-coordinate classical and relativistic geometries. Geometric-algebra elements (vectors) have dimension (grade), scalar amount (magnitude), space orientation/angle (direction), and relative direction {direction sense} (up or down, inside or outside, or positive or negative). Two vectors have a scalar dot product (inner product) {interior product}, bivector cross product (outer product), and inner product plus outer product (geometric product) (multivector). Grassmann and Clifford algebras generalize geometric algebra.
Hypernumber algebras {Clifford algebra} [1878] have 2^n dimensions for n number components. Dimensions represent reflections and rotations. Rotations are reflection combinations. Reflections convert right to left, or vice versa. Clifford algebras model spinors.
In generalized geometric algebras {Grassmann algebra}, the basis elements are the unit-magnitude dimensions, which can be any number and can be non-orthogonal. Elements are dimension linear combinations and have grade, magnitude, direction, and direction sense.
Operations are reflections. Elements add to make a new element. Elements multiply to make an element of one higher dimension (wedge product) {Grassmann product, algebra}. Parallel vectors are commutative. Perpendicular vectors are anti-commutative. Elements are associative for addition and multiplication. Grassmann algebra [1844 and 1862] is Clifford algebra in which two successive reflections cancel, rather than making rotation, and so there are no rotations and no need for metric or perpendicularity.
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Date Modified: 2022.0225