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.

`` The main bunch is outside, but there are some over there inside the wall ''.

Let me pass over the trip to Sante Fe with something of the same speed which made Mrs. Roebuck `` wonduh if the wahtahm speed limit '' ( 35 m.p.h. ) `` is still in ee-faket ''.

this is not so, for education offers all kinds of dividends, including how to pull the wool over a husband's eyes while you are having an affair with his wife.

`` That crap is softer over here ''.

Why, in the first place, call himself a liberal if he is against laissez-faire and favors an authoritarian central government with womb-to-tomb controls over everybody??

To him, law is the command of the sovereign ( the English monarch ) who personifies the power of the nation, while sovereignty is the power to make law -- i.e., to prevail over internal groups and to be free from the commands of other sovereigns in other nations.

It is the gait of the human who must run to live: arms dangling, legs barely swinging over the ground, head hung down and only occasionally swinging up to see the target, a loose motion that is just short of stumbling and yet is wonderfully graceful.

Nostalgic Yankee readers of Erskine Caldwell are today informed by proud Georgians that Tobacco Road is buried beneath a four-lane super highway, over which travel each day suburbanite businessmen more concerned with the Dow-Jones average than with the cotton crop.

Unruly hair goes straight up from his forehead, standing so high that the top falls gently over, as if to show that it really is hair and not bristle.

The information is furnished by each of the guests, is sent by oral broadcasting over the air waves, and is received by the ears.

The dweller at p is last to hear about a new cure, the slowest to announce to his neighbors his urgent distresses, the one who goes the farthest to trade, and the one with the greatest difficulty of all in putting over an idea or getting people to join him in a cooperative effort.

So in these pages the term `` technology '' is used to include any and all means which could amplify, project, or augment man's control over himself and over other men.

And the anxiety it generates is misinterpreted as anxiety over private interest and threatened social status.

The basic truth in the reactionary response is to be found in its realistic assumption of the primacy of the real over the ideational.

Going back over this ground and analyzing the composition of forces which have created the present scene is one of the tasks undertaken by the Center for the Study of Democratic Institutions, in Santa Barbara.

When I take over Taliesin, the first thing I'll do is fire you ''.

Carl thought the question over slowly and answered: `` I know a starving man who is fed never remembers all the pangs of his starvation, I know that ''.

It is, however, a disarming disguise, or perhaps a shield, for not only has Mercer proved himself to be one of the few great lyricists over the years, but also one who can function remarkably under pressure.

The `` conventional '' image of a particular time and place is not necessarily congruent with the image of the facts as established over the years by scholarly and scientific research.

For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing ( the more general ) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.

The two methods produce the same result ; string concatenation is associative ( but not commutative ).

If A itself is commutative ( as a ring ) then it is called a commutative R-algebra.

If A is commutative then the center of A is equal to A, so that a commutative R-algebra can be defined simply as a homomorphism of commutative rings.

The subcategory of commutative R-algebras can be characterized as the coslice category R / CRing where CRing is the category of commutative rings.

* Every polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra.

In fact, this is the free commutative R-algebra on the set

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order ( the axiom of commutativity ).

More compactly, an abelian group is a commutative group.

A group in which the group operation is not commutative is called a " non-abelian group " or " non-commutative group ".

This is not so obvious because xor is commutative and associative.