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2). Though Hermann Günther Grassmann (April 15, 1809–September 26, 1877) started the subject in 1844 with his Ausdehnungslehre (Theory of Extension)31 and re-published in 1862, his work was slow to find acceptance as ordinary linear algebra provided sufficient challenges to comprehension. After some preliminary work by Elwin Bruno Christoffel (November 10, 1829–March 15, 1900), a major advance in multilinear algebra came in the work of Gregorio Ricci-Curbastro (12 January 1853–6 August 1925) and Tullio Levi-Civita (29 March 1873–29 December 1941).
Geometric interpretation for the exterior product of k 1-forms (e, η, ω) to obtain a k-form (“mesh” of coordinate surfaces, here planes), for k = 1, 2, 3. The “circulations” show orientation. Algebra and the Geometric Interpretation of Vector Spaces • 35 An important type of multilinear forms is alternating multilinear forms which have the additional property of changing their sign under exchange of two arguments. The set of all alternating multilinear forms is a vector space, as the sum of two such maps, or the product of such a map with a scalar, is again alternating.
L. Doran argue that it provides compact and intuitive descriptions in many areas, including classical and quantum mechanics, electromagnetic theory and relativity. 57 Clifford algebras are a type of associative algebra. They are named after the English geometer William Kingdon Clifford (4 May 1845–3 March 1879). As K -algebras (an algebra over a field K, where K is called the base field of A), Clifford algebras generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems.