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 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 )
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