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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 algebra generated by the geometric product (that is, all objects formed by taking repeated sums and geometric products of scalars and vectors) is the geometric algebra over the vector space. For an Euclidean vector space, this algebra is written G n {\displaystyle {\mathcal {G}}_{n}} or Cl n ( R ) , where n is the dimension of the vector ...
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems. Classically, it studies zeros of multivariate polynomials ; the modern approach generalizes this in a few different aspects.
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 ]
A Clifford algebra is a unital associative algebra that contains and is generated by a vector space V over a field K, where V is equipped with a quadratic form Q : V → K.The Clifford algebra Cl(V, Q) is the "freest" unital associative algebra generated by V subject to the condition [c] = , where the product on the left is that of the algebra, and the 1 on the right is the algebra's ...
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.
GeoGebra is software that combines geometry, algebra and calculus for mathematics education in schools and universities. It is available free of charge for non-commercial users. [6] License: open source under GPL license (free of charge) Languages: 55; Geometry: points, lines, all conic sections, vectors, parametric curves, locus lines
In multilinear algebra, a multivector, sometimes called Clifford number or multor, [1] is an element of the exterior algebra Λ(V) of a vector space V.This algebra is graded, associative and alternating, and consists of linear combinations of simple k-vectors [2] (also known as decomposable k-vectors [3] or k-blades) of the form
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