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| An '''algebra over a field''' (or simply an '''algebra''') is a ] ''A'' together with a vector multiplication that distributes over vector addition and has the further property that (''ax'')(''by'') = (''ab'')(''xy'') for all scalars ''a'' and ''b'' and all vectors ''x'' and ''y''. Such a vector multiplication is a ] ''A'' x ''A'' <tt>-></tt> ''A''. The most important types of algebras are the ], such as algebras of ] or ], and the ], such as '''R'''<sup>3</sup> with the multiplication given by the ] or algebras of ]. | An '''algebra over a field''' (or simply an '''algebra''') is a ] ''A'' together with a vector multiplication that distributes over vector addition and has the further property that (''ax'')(''by'') = (''ab'')(''xy'') for all scalars ''a'' and ''b'' and all vectors ''x'' and ''y''. Such a vector multiplication is a ] ''A'' x ''A'' <tt>-></tt> ''A''. The most important types of algebras are the ], such as algebras of ] or ], and the ], such as '''R'''<sup>3</sup> with the multiplication given by the ] or algebras of ]. | ||
| Other examples are the ] and the ]. | |||
| See also ], ] and ]. | See also ], ] and ]. | ||
Revision as of 17:14, 9 January 2002
The term algebra is used in mathematics in several different senses.
At an elementary level, algebra involves the manipulation of simple equations in real (or sometimes complex) variables. See Elementary algebra.
More generally, algebra (or abstract algebra) is the study of algebraic structures such as groups, rings and fields. See Abstract algebra for further details.
An algebra over a field (or simply an algebra) is a vector space A together with a vector multiplication that distributes over vector addition and has the further property that (ax)(by) = (ab)(xy) for all scalars a and b and all vectors x and y. Such a vector multiplication is a bilinear map A x A -> A. The most important types of algebras are the associative algebras, such as algebras of matrices or polynomials, and the Lie algebras, such as R with the multiplication given by the vector cross product or algebras of vector fields.
Other examples are the octonions and the sedenions.
See also Boolean algebra, sigma-algebra and linear algebra.