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Meaning of ALGEBRA

Pronunciation:  'aljubru

 
WordNet Dictionary
 
 Definition: 
[n]  the mathematics of generalized arithmetical operations
 

ALGEBRA is a 7 letter word that starts with A.

 

 See Also: linear algebra, matrix algebra, pure mathematics, quadratics, vector algebra

 

 

Webster's 1913 Dictionary
 
 Definition: 
\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.

2. A treatise on this science.

 
Computing Dictionary
 
 Definition: 

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"].

 

 

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