# Algebras

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1. A loose term for anÂ algebraic structure.2. AÂ vector spaceÂ that is also aÂ 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 2x2Â matricesÂ withÂ real numbersÂ as entries, with the usual operations of addition and matrix multiplication, and the usualÂ scalarÂ multiplication. another example is the set of allÂ polynomialsÂ with real coefficients, with the usual operations.In more detail, we have:(1) an underlyingÂ set,(2) aÂ fieldÂ ofÂ 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 aÂ vector space,(4) an operation of addition of members of the underlying set, whose input is anÂ ordered pairÂ 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 aÂ Banach algebra) is the set of allÂ boundedÂ linear operatorsÂ on aÂ Hilbert space, with the usualÂ norm. the multiplication is the operation ofÂ compositionÂ of operators, and the addition and scalar multiplication are just what you would expect.Two other examples areÂ tensor algebrasÂ andÂ Clifford algebras.[I. N. Herstein, "Topics_in_Algebra"].(1999-07-14)

Algebra is a branch ofÂ mathematicsÂ concerning the study ofÂ structure,Â relationÂ andÂ quantity. the name is derived from the treatise written by theÂ ArabicÂ mathematician,Â astronomer,Â astrologerÂ andÂ geographer, titled (inÂ ArabicÂ Ã˜Â§Ã™Â„Ã™ÂƒÃ˜ÂªÃ˜Â§Ã˜Â¨ Ã˜Â§Ã™Â„Ã˜Â¬Ã˜Â¨Ã˜Â± Ã™ÂˆÃ˜Â§Ã™Â„Ã™Â…Ã™Â‚Ã˜Â§Ã˜Â¨Ã™Â„Ã˜Â© ) 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 ofÂ linearÂ andÂ quadratic equations.

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

• aalgebraas
• aelgebraes
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### Typos

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#### Word analysis of algebras

 Length 8 Vocals algebras Consonants algebras MD5 a1e445cf27478121b04341b7a7f1ac45 SHA1 2a040fc37c21ad11dab75f6dd3f48416e02678e1