| Revision as of 15:43, 25 February 2002 editZundark (talk | contribs)Extended confirmed users, File movers, Pending changes reviewers29,657 edits algebra multiplication completely determined by multiplication of basis elements← Previous edit | Revision as of 14:23, 9 April 2002 edit undoBryan Derksen (talk | contribs)Extended confirmed users95,333 editsmNo edit summaryNext edit → | ||
| Line 3: | Line 3: | ||
| At an elementary level, '''algebra''' involves the manipulation of simple ] in ] (or sometimes ]) variables. See ]. | At an elementary level, '''algebra''' involves the manipulation of simple ] in ] (or sometimes ]) variables. See ]. | ||
| More generally, '''algebra''' (or '''abstract algebra''') is the study of algebraic structures such as ], ] and ]. See ] for further details. | More generally, '''algebra''' (or '''abstract algebra''') is the study of algebraic structures such as ], ] and ]. See ] for further details. | ||
| 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''. | 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''. | ||
Revision as of 14:23, 9 April 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, and is therefore completely determined by the multiplication of basis elements of 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.