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In mathematics, a geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors ...
The bivectors of the three-dimensional Euclidean space form the Lie algebra, which is isomorphic to the () Lie algebra. This accidental isomorphism allows to picture a geometric interpretation of the states of the two dimensional Hilbert space by using the Bloch sphere. One of those systems is the spin 1/2 particle.
Plane-based geometric algebra is an application of Clifford algebra to modelling planes, lines, points, and rigid transformations. Generally this is with the goal of solving applied problems involving these elements and their intersections , projections , and their angle from one another in 3D space. [ 1 ]
In mathematics, geometric calculus extends geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to reproduce other mathematical theories including vector calculus , differential geometry , and differential forms .
In geometric algebra, the outermorphism of a linear function between vector spaces is a natural extension of the map to arbitrary multivectors. [1] It is the unique unital algebra homomorphism of exterior algebras whose restriction to the vector spaces is the original function.
In this article, certain applications of the dual quaternion algebra to 2D geometry are discussed. At this present time, the article is focused on a 4-dimensional subalgebra of the dual quaternions which will later be called the planar quaternions. The planar quaternions make up a four-dimensional algebra over the real numbers.
Similarly, Fermat's Last Theorem is stated in terms of elementary arithmetic, which is a part of commutative algebra, but its proof involves deep results of both algebraic number theory and algebraic geometry. Noncommutative rings are quite different in flavour, since more unusual behavior can arise.
Quadric geometric algebra (QGA) is a geometrical application of the , geometric algebra.This algebra is also known as the , Clifford algebra.QGA is a super-algebra over , conformal geometric algebra (CGA) and , spacetime algebra (STA), which can each be defined within sub-algebras of QGA.
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