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It is a vector space over the residue field k := R / m.
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is and vector
It is clear that each vector in the range of Af is in Af for if **ya is in the range of Af, then Af and so Af because Af is divisible by the minimal polynomial P.
From these results, one sees that the study of linear operators on vector spaces over an algebraically closed field is essentially reduced to the study of nilpotent operators.
In the primary decomposition theorem, it is not necessary that the vector space V be finite dimensional, nor is it necessary for parts ( A ) and ( B ) that P be the minimal polynomial for T.
If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.
If the rake angle **yc of the knife is high enough and the friction angle **yt between the front of the knife and the back of the chip is low enough to give a positive value for Af, the resultant vector R will lie above the plane of the substrate.
If T is the total `` length '' of the process, its feed state may be denoted by a vector p(T) and the product state by p(Q).
The Asteroids arcade machine is a vector game.
Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact.
** On every infinite-dimensional topological vector space there is a discontinuous linear map.
For example an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces.
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 ).
is and space
Experience is not seen, as it is in classical rationalism, as presenting us initially with clear and distinct objects simply located in space and registering their character, movements, and changes on the tabula rasa of an uninvolved intellect.
It is notable that at this time he was writing with admiration of Cimabue's and Poussin's way of filling space.
Next I refer to our program in space exploration, which is often mistakenly supposed to be an integral part of defense research and development.
Third, the United States is pressing forward in the development of large rocket engines to place vehicles of many tons into space for exploration purposes.
Like ours, the economy of the space merchants must constantly expand in order to survive, and, like ours, it is based on the principle of `` ever increasing everybody's work and profits in the circle of consumption ''.
And this, of course, is exactly what Madison Avenue has been accused of doing albeit in a primitive way, with its `` hidden persuaders '' and what the space merchants accomplish with much greater sophistication and precision.
Thus the copywriter in the world of the space merchants is the person who in earlier ages might have been a lyric poet, the person `` capable of putting together words that stir and move and sing ''.
But there is hope, for Conservation Commissioner Bontempo has tagged the sanctuary as the kind of place the state hopes to include in its program to double its park space.
Four hundred million dollars of the increase is for the expanded space program, a responsibility similarly neglected by Mr. Eisenhower.
I would like to see you devote some space in an early issue to the news blackout concerning President Kennedy's activities, so far as Southern California is concerned.
Advice is given also on problems of plant location and plant space.
The outside 4-inch space is filled by mortaring blocks on edge.
The space between them is filled with pit-run gravel or earth.
In general, such apartments afford more protection than smaller buildings because their walls are thick and there is more space.
The problem for the city apartment dweller is primarily to plan the use of existing space.
The music is always allowed the living space needed to attain its full sonority ; ;
But if space and money are no problem and small children are not on hand every day, it is certainly more restful to have your pool and entertainment area removed from the immediate environs of the house.
Since there is a continual loss of micrometeoritic material in space because of the radiation effects, there must be a continual replenishment: otherwise, micrometeorites would have disappeared from interplanetary space.
With detectors sensitive to three mass intervals and based on a few counts, the second and third Russian space probes indicate that the flux of the smallest particles detected is less than that of larger ones.
Of course, if there is a dust blanket around the Earth, the fluxes in interplanetary space should be less than the figures given here.
If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.
Conversely, suppose that **ya is in the null space of Af.
is and over
`` 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.
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