We have chosen to give it at the end of the section since it deals with differential equations and thus is not purely linear algebra.

has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.

* In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL ( V ).

The same definition holds in any unital ring or algebra where a is any invertible element.

In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry.

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

His notion of abelian category is now the basic object of study in homological algebra.

With the existence of an alpha channel, it is possible to express compositing image operations, using a compositing algebra.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

Explicitly, is an associative algebra homomorphism if

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* An example of a non-unitary associative algebra is given by the set of all functions f: R → R whose limit as x nears infinity is zero.

* Any ring of characteristic n is a ( Z / nZ )- algebra in the same way.

* Any ring A is an algebra over its center Z ( A ), or over any subring of its center.

* Any commutative ring R is an algebra over itself, or any subring of R.

Other important Arabic astrologers include Albumasur and Al Khwarizmi, the Persian mathematician, astronomer and astrologer, who is considered the father of algebra and the algorithm.

My sincere wish is that he continues to add to this record he sets here today.

In the electronics industry, this tendency is well illustrated by inventories of TV sets.

The last exercise of Roland Claude's prescribed program for Henri is a single exercise, done in individual sets with a bit longer pause between sets.

It is the one exercise that drastically influences the definition of the thighs at the hipline -- that mark of the champion that sets him apart from all other bodybuilders ; ;

The greatest difference in the two sets of figures is due to differences in the two sets of lists used.

Thus, if public pressure sets the effective limit to the price that the industry may charge, this pressure is itself a function of the wage rate.

This in turn sets off an alarm, notifying the observers at the station that a tsunami is in progress.

The mandate is distinguished from the appeal court's opinion, which sets out the legal reasoning for its decision.

But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols.

There is a wide variety of representations possible and one can express a given Turing machine program as a sequence of machine tables ( see more at finite state machine, state transition table and control table ), as flowcharts ( see more at state diagram ), or as a form of rudimentary machine code or assembly code called " sets of quadruples " ( see more at Turing machine ).

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".

More explicitly, it is stating that for every indexed family of nonempty sets there exists an indexed family of elements such that for every.

A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f ( s ) is an element of s. With this concept, the axiom can be stated:

Each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X.

This is not the most general situation of a Cartesian product of a family of sets, where a same set can occur more than once as a factor ; however, one can focus on elements of such a product that select the same element every time a given set appears as factor, and such elements correspond to an element of the Cartesian product of all distinct sets in the family.

: Given any family of nonempty sets, their Cartesian product is a nonempty set.

Its domain is the powerset of A ( with the empty set removed ), and so makes sense for any set A, whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets.

After the Griffin-Byrd political troup has completed the circuit in November in the name of a Pre-Legislative Forum, this is going to be the most politically oriented Legislature in history.

Without such forward planning, investment funds are wasted because manufacturing facilities are completed before there is power to operate them or before there is transport to service them ; ;

The basic mapping phase of the program has been completed and the inventory phase is scheduled for completion July 1, 1961.

As soon as the shelter is completed a radio reception check must be made.

Much of this necessary increase in research and development, though properly chargeable to current expenses, is not reflected in earnings until projects are completed and the new machines sold in quantity, usually over a period of several years.

When this linear draft is completed, I dust it down to a faint image.

These maladjustments in turn inhibit learning, and a vicious cycle is completed.

In the period from 10 to 14 the permanent set of teeth is usually completed, yet the continuing growth of bony tissue makes moving badly placed teeth comparatively easy.

The conversion to magnetic tape is not yet completed, he said, and added Field's long service in state government and welfare employ gave him familiarity with the welfare program.

It is the AID's intention to create in the library `` a miniature museum of Americana '' before completed refurbishing is unveiled early this fall.

The work of Lay Visitation Evangelism is not completed when all of the persons on the Responsibility List have been interviewed.

But now the task is completed and the uncertainty resolved with the opening of the English-dialogue picture at the Music Hall yesterday.

The New York Shakespeare Festival, which is using the Wollman Memorial Skating Rink while its theatre near the Belvedere is being completed, began bravely.

The form is completed by the appellant or by the appellant's legal representative.

One Nepōhualtzintzin ( 91 ) represented the number of days that a season of the year lasts, two Nepōhualtzitzin ( 182 ) is the number of days of the corn's cycle, from its sowing to its harvest, three Nepōhualtzintzin ( 273 ) is the number of days of a baby's gestation, and four Nepōhualtzintzin ( 364 ) completed a cycle and approximate a year ( 1 days short ).

Reconstruction of these three lines began in 2005 and is expected to be completed by the end of the year 2012.

The road between Lubango and Namibe, for example, was completed recently with funding from the European Union, and is comparable to many European main routes.

It is followed by the Nihon shoki, completed in 720, and that by the Man ' yōshū, which dates from c. 771-785, but includes material that is about 400 years earlier ( Miller 1971: 4 ).

* 1998 – Travelers Group announces an agreement to undertake the $ 76 billion merger between Travelers and Citicorp, and the merger is completed on October 8, of that year, forming Citibank.

* 1993 – The Rainbow Bridge, connecting Tokyo's Shibaura and the island of Odaiba, is completed.