1. FINITE-DIMENSIONAL VECTOR SPACES
... By now you’ll have acquired a fair knowledge of matrices. These are a concrete embodiment of something rather more abstract. Sometimes it is easier to use matrices, but at other times the abstract approach allows us more freedom. Recall that a field is a mathematical system having two operations + a ...
... By now you’ll have acquired a fair knowledge of matrices. These are a concrete embodiment of something rather more abstract. Sometimes it is easier to use matrices, but at other times the abstract approach allows us more freedom. Recall that a field is a mathematical system having two operations + a ...
Document
... Dot product is always 0 between perpendicular vectors If V and W are unit vectors, the dot product is 1 for parallel vectors pointing in the same direction, -1 for opposite ...
... Dot product is always 0 between perpendicular vectors If V and W are unit vectors, the dot product is 1 for parallel vectors pointing in the same direction, -1 for opposite ...
MATH 412: NOTE ON INFINITE-DIMENSIONAL VECTOR SPACES
... we choose the answer to be “yes”. 2. General definitions We now summarize what we saw in the examples above. For this let V be a vector space over a field F , and fix S ⊂ V . The proofs are all the same as in the finite-dimensional case and can also be found in my notes for Math 223 (see my website) ...
... we choose the answer to be “yes”. 2. General definitions We now summarize what we saw in the examples above. For this let V be a vector space over a field F , and fix S ⊂ V . The proofs are all the same as in the finite-dimensional case and can also be found in my notes for Math 223 (see my website) ...
Geometric algebra
A geometric algebra (GA) is a Clifford algebra of a vector space over the field of real numbers endowed with a quadratic form. The term is also sometimes used as a collective term for the approach to classical, computational and relativistic geometry that applies these algebras. The Clifford multiplication that defines the GA as a unital ring is called the geometric product. Taking the geometric product among vectors can yield bivectors, trivectors, or general n-vectors. The addition operation combines these into general multivectors, which are the elements of the ring. This includes, among other possibilities, a well-defined formal sum of a scalar and a vector.Geometric algebra is distinguished from Clifford algebra in general by its restriction to real numbers and its emphasis on its geometric interpretation and physical applications. Specific examples of geometric algebras applied in physics include the algebra of physical space, the spacetime algebra, and the conformal geometric algebra. Geometric calculus, an extension of GA that incorporates differentiation and integration can be used to formulate other theories such as complex analysis, differential geometry, e.g. by using the Clifford algebra instead of differential forms. Geometric algebra has been advocated, most notably by David Hestenes and Chris Doran, as the preferred mathematical framework for physics. Proponents claim that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity. GA has also found use as a computational tool in computer graphics and robotics.The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra, which is the geometric algebra of the trivial quadratic form. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them ""geometric algebras""). For several decades, geometric algebras went somewhat ignored, greatly eclipsed by the vector calculus then newly developed to describe electromagnetism. The term ""geometric algebra"" was repopularized by Hestenes in the 1960s, who recognized its importance to relativistic physics.