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An algebra over a field (or simply an algebra) is a vector space 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. For example, a field is an algebra over any of its subfields, and the quaternions, octonions and sedenions are algebras over the real numbers. Another example is R3 with the usual 3-dimensional vector multiplication. |
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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 R3 with the multiplication given by the vector cross product or algebras of [vector fields]?. |
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See also Boolean algebra, sigma-algebra? and linear algebra. |
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See also Boolean algebra, sigma-algebra? and linear algebra. |
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 R3 with the multiplication given by the vector cross product or algebras of [vector fields]?.
See also Boolean algebra, sigma-algebra? and linear algebra.
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