A geometric algebra is the Clifford algebra of a vector space over the field of real numbers endowed with a quadratic form.

Given a finite dimensional real quadratic space with quadratic form, the geometric algebra for this quadratic space is the Clifford algebra Cℓ ( V, Q ).

Moreover, the algebra generated by the three matrices is isomorphic to the 3-dimensional Euclidean real Clifford Algebra.

In detail, if V is a finite-dimensional complex vector space with nondegenerate bilinear form g, the Clifford algebra is the algebra generated by V along with the anticommutation relation.

The Clifford algebra Cℓ < sub > n </ sub >( C ) is algebraically isomorphic to the algebra of complex matrices, if is even ; or the algebra of two copies of the matrices, if is odd.

The Lie algebra is embedded as a Lie subalgebra in equipped with the Clifford algebra commutator as Lie bracket.

Irreducible representations over the reals in the case when V is a real vector space are much more intricate, and the reader is referred to the Clifford algebra article for more details.

The most typical type of spinor, the Dirac spinor, is an element of the fundamental representation of the complexified Clifford algebra, into which the spin group Spin ( p, q ) may be embedded.

Some simple examples of spinors in low dimensions arise from considering the even-graded subalgebras of the Clifford algebra.

As a real algebra, Cℓ < sup > 0 </ sup >< sub > 2, 0 </ sub >( R ) is isomorphic to field of complex numbers C. As a result, it admits a conjugation operation ( analogous to complex conjugation ), sometimes called the reverse of a Clifford element, defined by

The Clifford algebra Cℓ < sub > 3, 0 </ sub >( R ) is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, σ < sub > 1 </ sub >, σ < sub > 2 </ sub > and σ < sub > 3 </ sub >, the three unit bivectors σ < sub > 1 </ sub > σ < sub > 2 </ sub >, σ < sub > 2 </ sub > σ < sub > 3 </ sub >, σ < sub > 3 </ sub > σ < sub > 1 </ sub > and the pseudoscalar i

equivalence of these constructions are a consequence of the uniqueness of the spinor representation of the complex Clifford algebra.

Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows.

Then the assignment extends uniquely to an algebra homomorphism by sending the monomial in the Clifford algebra to the product of matrices and extending linearly.

In this construction the representation of the Clifford algebra Cℓ ( V, g ), the Lie algebra so ( V, g ), and the Spin group Spin ( V, g ), all depend on the choice of the orthonormal basis and the choice of the gamma matrices.

These are subspaces of the Clifford algebra of the form Cℓ ( V, g ) ω, admitting the evident action of Cℓ ( V, g ) by left-multiplication: c: xω → cxω.

The action of an even Clifford element on vectors, regarded as 1-graded elements of Cℓ < sub > 2, 0 </ sub >, is determined by mapping a general vector to the vector

Cℓ < sup > odd </ sup > ω on restriction to the action of the even Clifford algebra.

The Clifford algebra Cℓ ( V, Q ) is the " freest " algebra generated by V subject to the condition < ref > Mathematicians who work with real Clifford algebras and prefer positive definite quadratic forms ( especially those working in index theory ) sometimes use a different choice of sign in the fundamental Clifford identity.

In fact, if then the Clifford algebra Cℓ ( V, Q ) is just the exterior algebra Λ ( V ).

A Clifford algebra Cℓ ( V, Q ) is a unital associative algebra over K together with a linear map satisfying for all defined by the following universal property: Given any associative algebra A over K and any linear map such that

It is then straightforward to show that Cℓ ( V, Q ) contains V and satisfies the above universal property, so that Cℓ is unique up to a unique isomorphism ; thus one speaks of " the " Clifford algebra Cℓ ( V, Q ).

The universal characterization of the Clifford algebra shows that the construction of Cℓ ( V, Q ) is functorial in nature.

In mathematics, the split-complex numbers are members of the Clifford algebra Cℓ < sub > 1, 0 </ sub >( R ) = Cℓ < sup > 0 </ sup >< sub > 1, 1 </ sub >( R ) ( the superscript 0 indicating the even subalgebra ).

Below it is shown that distinct Clifford algebras may be algebra isomorphic, as is the case of Cℓ < sub > 2, 0 </ sub >( R ) and Cℓ < sub > 1, 1 </ sub >( R ) which are both isomorphic to the ring of two-by-two matrices over the real numbers.

We will denote the Clifford algebra on C < sup > n </ sup > with the standard quadratic form by Cℓ < sub > n </ sub >( C ).

In physics, the algebra of physical space ( APS ) is the use of the Clifford or geometric algebra Cℓ < sub > 3 </ sub > of the three-dimensional Euclidean space as a model for ( 3 + 1 )- dimensional space-time, representing a point in space-time via a paravector ( 3-dimensional vector plus a 1-dimensional scalar ).

* Clifford Stoll: < cite > Cuckoo's Egg: Tracking a Spy Through the Maze of Computer Espionage </ cite >, Pocket Books, ISBN 0-7434-1146-3

It therefore has a unique irreducible representation ( also called simple Clifford module ), commonly denoted by Δ, whose dimension is 2 < sup > k </ sup >.

where γ < sup >*</ sup > is the conjugate of γ, and the product is Clifford multiplication.

) In 1878, William Kingdon Clifford joined this exterior algebra to William Rowan Hamilton's quaternions by replacing Grassmann's rule e < sub > p </ sub > e < sub > p </ sub >

The editors of this volume and Volume 2, were the late Charles Palache, Clifford Frondel, and the late Harry Berman, all of Harvard University.

Quadratic forms and Clifford algebras in characteristic 2 form an exceptional case.

On the night of 10 June, at Bramham Moor, outside Tadcaster, Lovell led 2, 000 men on a night attack against 400 Lancastrians, led by Lord Clifford.

Clifford continued McNamara's highly publicized Cost Reduction Program, announcing that over $ 1. 2 billion had been saved in fiscal year ( FY ) 1968 as a result of the effort.

* Ward 2: Dr. Clifford Keezer ( 2011 ) and Paul Nevadomski ( 2012 )

For example, in Act 2, Scene 1, it is Clarence who says Edward's " I wonder how our princely father scaped ,/ Or whether he be scaped away or no / From Clifford and Northumberland's pursuit " ( ll. 1 – 3 ).

For example, in Act 2, Scene 1, it is Clarence who says Edward's " I wonder how our princely father scaped ,/ Or whether he be scaped away or no / From Clifford and Northumberland's pursuit ".

Cruz had two children ; Carlos, Jr. ( born 1962 ) who has four children, Bradely Cruz ( born 1990 ) Clifford Allen Cruz & Clifton Allen Cruz ( born 1992 ), Brandon Miguel Cruz ( born 1996 ), and Hermina ( born 1963 ) having 2 children Alexis Tatia Cruz ( born 1998 ) and Bryant Lope Cruz

Clifford Everett " Bud " Shank, Jr. ( May 27, 1926 – April 2, 2009 ) was an American alto saxophonist and flautist.

His biggest hits, " Honky Tonk " ( the Part 2 side of the record ) and " Slow Walk " featured saxophonist Clifford Scott.

Lucy Clifford ( 2 August 1846 – 21 April 1929 ), better known as Mrs. W. K. Clifford, was a British novelist and journalist, and the wife of William Kingdon Clifford.

* Sir Clifford Campbell ( 1 December 1962 – 2 March 1973 )

On flight day two, the Red Team crew of Commander Gutierrez, Pilot Chilton and Linda M. Godwin began its sleep shift about 5 pm Central time, to awaken at 2 am The Blue Team crew members, Jay Apt, Michael R. Clifford and Thomas D. Jones, awakened about 4 pm to begin their third flight day on orbit, and would go to bed about 5 am

Clifford first flew on the crew of STS-53 which launched from the Kennedy Space Center, Florida, on December 2, 1992, aboard the Space Shuttle Discovery.

Mathematical physicists James Cockle, William Kingdon Clifford, and Alexander Macfarlane had all employed in their writings an equivalent mapping of the Cartesian plane by operator ( cosh a + r sinh a ), where a is the hyperbolic angle and r < sup > 2 </ sup > = + 1.