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Definition by Wiktionary (Licensed under Creative Commons Attribution/Share-Alike License)

1. A loose term for an algebraic structure.2. A vector space that is also a ring, where the vector space and the ring share the same addition operation and are related in certain other ways.An example algebra is the set of 2x2 matrices with real numbers as entries, with the usual operations of addition and matrix multiplication, and the usual scalar multiplication. another example is the set of all polynomials with real coefficients, with the usual operations.In more detail, we have:(1) an underlying set,(2) a field of scalars,(3) an operation of scalar multiplication, whose input is a scalar and a member of the underlying set and whose output is a member of the underlying set, just as in a vector space,(4) an operation of addition of members of the underlying set, whose input is an ordered pair of such members and whose output is one such member, just as in a vector space or a ring,(5) an operation of multiplication of members of the underlying set, whose input is an ordered pair of such members and whose output is one such member, just as in a ring.This whole thing constitutes an `algebra' iff:(1) it is a vector space if you discard item (5) and(2) it is a ring if you discard (2) and (3) and(3) for any scalar r and any two members A, B of the underlying set we have r(AB) = (rA)B = A(rB). In other words it doesn't matter whether you multiply members of the algebra first and then multiply by the scalar, or multiply one of them by the scalar first and then multiply the two members of the algebra. note that the A comes before the B because the multiplication is in some cases not commutative, e.g. the matrix example.Another example (an example of a Banach algebra) is the set of all bounded linear operators on a Hilbert space, with the usual norm. the multiplication is the operation of composition of operators, and the addition and scalar multiplication are just what you would expect.Two other examples are tensor algebras and Clifford algebras.[I. N. Herstein, "Topics_in_Algebra"].(1999-07-14)

Algebra is a branch of mathematics concerning the study of structure, relation and quantity. the name is derived from the treatise written by the Arabic mathematician, astronomer, astrologer and geographer, titled (in Arabic Ø§Ù„كتاب الجبر والمقابلة ) Al-Kitab al-Jabr wa-l-Muqabala (meaning "The compendious book on calculation by completion and Balancing"), which provided symbolic operations for the systematic solution of linear and quadratic equations.

Noun1. the mathematics of generalized arithmetical operations (hypernym) pure mathematics (hyponym) quadratics (classification) mathematics, math, maths (class) transposition

Common misspellings

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Typos

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Anagrams

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Word analysis of algebras

Length8
Vocalsalgebras
Consonantsalgebras
MD5a1e445cf27478121b04341b7a7f1ac45
SHA12a040fc37c21ad11dab75f6dd3f48416e02678e1