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algebra 音标拼音: ['ældʒəbrə] n. 代数学 代数学 algebra 代数 algebra n 1: the mathematics of generalized arithmetical operations Mathematics \Math`e*mat"ics\, n. [F. math['e]matiques, pl., L. mathematica, sing., Gr. ? (sc. ?) science. See {Mathematic}, and {-ics}.] That science, or class of sciences, which treats of the exact relations existing between quantities or magnitudes, and of the methods by which, in accordance with these relations, quantities sought are deducible from other quantities known or supposed; the science of spatial and quantitative relations. [1913 Webster] Note: Mathematics embraces three departments, namely: 1. {Arithmetic}. 2. {Geometry}, including {Trigonometry} and {Conic Sections}. 3. {Analysis}, in which letters are used, including {Algebra}, {Analytical Geometry}, and {Calculus}. Each of these divisions is divided into pure or abstract, which considers magnitude or quantity abstractly, without relation to matter; and mixed or applied, which treats of magnitude as subsisting in material bodies, and is consequently interwoven with physical considerations. [1913 Webster] Algebra \Al"ge*bra\, n. [LL. algebra, fr. Ar. al-jebr reduction of parts to a whole, or fractions to whole numbers, fr. jabara to bind together, consolidate; al-jebr w'almuq[=a]balah reduction and comparison (by equations): cf. F. alg[`e]bre, It. & Sp. algebra.] 1. (Math.) That branch of mathematics which treats of the relations and properties of quantity by means of letters and other symbols. It is applicable to those relations that are true of every kind of magnitude. [1913 Webster] 2. A treatise on this science. [1913 Webster] 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[Noun]
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