algebra
n. mathematical system that uses equations containing letters and numbers | ||||
Search Dictionary:
Algebra definition was found in categories: Computer & Internet(1) Language, Idioms & Slang(5) Arts & Humanities(1) Entertainment & Music(1) Encyclopedia(1)
Algebra Definition from Computer & Internet Dictionaries & Glossaries
| FOLDOC |
algebra
<mathematics, logic> 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)
<mathematics, logic> 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 Definition from Language, Idioms & Slang Dictionaries & Glossaries
| Webster's Revised Unabridged Dictionary (1913) |
Algebra
(n.)
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.
(n.)
A treatise on this science.
(n.)
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.
(n.)
A treatise on this science.
| WordNet 2.0 |
algebra
Noun
1. the mathematics of generalized arithmetical operations
(hypernym) pure mathematics
(hyponym) quadratics
(classification) mathematics, math, maths
(class) transposition
Noun
1. the mathematics of generalized arithmetical operations
(hypernym) pure mathematics
(hyponym) quadratics
(classification) mathematics, math, maths
(class) transposition
| hEnglish - advanced version |
algebra
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'almuqābalah reduction and comparison (by equations): cf. f. algèbre, 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. [
algebra
n : the mathematics of generalized arithmetical operations
similar words(9)
banach algebra
information algebra
relational algebra
boolean algebra
universal algebra
graphic algebra
multiple algebra
linear algebra
vector algebra
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'almuqābalah reduction and comparison (by equations): cf. f. algèbre, 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. [
algebra
n : the mathematics of generalized arithmetical operations
similar words(9)
banach algebra
information algebra
relational algebra
boolean algebra
universal algebra
graphic algebra
multiple algebra
linear algebra
vector algebra
| Concise English-Irish Dictionary v. 1.1 |
algebra
ailgéabar
ailgéabar
| JM Welsh <=> English Dictionary |
Alsawdd
Alsawdd = n. algebra
Alsawdd = n. algebra
Algebra Definition from Arts & Humanities Dictionaries & Glossaries
| JM Latin-English Dictionary |
algebra
N F
algebra
N F
algebra
Algebra Definition from Entertainment & Music Dictionaries & Glossaries
| English to Federation-Standard Golic Vulcan |
Algebra
seht-su'us-ek'tal
seht-su'us-ek'tal
Algebra Definition from Encyclopedia Dictionaries & Glossaries
| Wikipedia English - The Free Encyclopedia |
Algebra
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 الكتاب 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.
| See more at Wikipedia.org... |
*-algebra
*-ring
In mathematics, a *-ring is an associative ring with a map * : A → A which is an antiautomorphism, and an involution.More precisely, * is required to satisfy the following properties:
This also called an involutive ring, involutory ring, and ring with involution.
| See more at Wikipedia.org... |