For vector spaces over non-algebraically closed fields, we still need to find some substitute for characteristic values and vectors.

For each picture frame, the 6502 writes graphics commands for the DVG into a defined area of RAM ( the vector RAM ), and then asks the DVG to draw the corresponding vector image on the screen.

For example an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces.

For an object with a fixed mass that is rotating about a fixed symmetry axis, the angular momentum is expressed as the product of the moment of inertia of the object and its angular velocity vector:

For any system, the following restrictions on measurement results apply, where is the reduced Planck constant and is any direction vector such as x, y, or z:

For a finite-dimensional vector space, using a fixed orthonormal basis, the inner product can be written as a matrix multiplication of a row vector with a column vector:

For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication:

For 2D vector animations, the rendering process is the key frame illustration process, while tweened frames are rendered as needed.

For example, binary string 1011 is represented by the vector ( 1, 0, 1, 1 ).

For the purposes of this article, we call this constructed vector the transmitted vector.

For a vector expressed in cylindrical coordinates as

For twisting rotation to exist, it is necessary to tilt the body sideways after takeoff, so that there is now a small component of this horizontal angular momentum vector along the body's long axis.

Malaria Foundation International states, " The outcome of the treaty is arguably better than the status quo going into the negotiations ... For the first time, there is now an insecticide which is restricted to vector control only, meaning that the selection of resistant mosquitoes will be slower than before.

For the special case, this implies that the length of a vector is preserved as well — this is just Parseval's theorem:

For a flow, the vector field Φ ( x ) is a linear function of the position in the phase space, that is,

For a more useful solution, we utilize vector identities, which work for any vector, as follows:

For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about endomorphisms in any category.

For any positive integer n, the set of all n-tuples of real numbers forms an n-dimensional vector space over R, which is denoted R < sup > n </ sup > and sometimes called real coordinate space.

For example, a bijective linear map is an isomorphism between vector spaces, and a bijective continuous function whose inverse is also continuous is an isomorphism between topological spaces, called a homeomorphism.

For, with a single exception, nothing had happened to them.

For Tom Horn, it turned out, had a number of rancher and cowboy witnesses ready and willing to swear with straight faces that he had been in Bates Hole the day of the killing.

For a brief period each year, the rays of the sun are warm enough to melt some of the snows piled a mile deep at the base of the headwalls, and then the pinnacles glisten in the daytime at high noon, and billions of gallons of water begin their slow seepage under the glaciers and across the rockstrewn hanging valleys on their long, meandering journey to the sea -- running east past the sky-carving massifs of Gurla Mandhata and Kemchenjunga, then turning south and curling down through the jungles of Assam, past the Khasi Hills, and into Bengal, past Sirinjani and Madaripur, until the hard water of the melting snows mingles with the soft drainage of fields and at length fans out to meld with the teeming salt depths of the Bay of Bengal.

For several months now, Jack Carter, a big overgrown boy of fifteen with a fuzzy, pimpled face and greenish catlike eyes with a lot of red in them, had been haunted by a dream, a vision, of a Woman.

For Mr. Taylor's Images And Reflections she made some diaphanous tents that alternately hide and reveal the performer, and a girl's cape lined with grass.

For the family is the simplest example of just such a unit, composed of people, which gives us both some immunity from, and a way of dealing with, other people.

For several generations much fiction has appeared dealing with the steprelationship.

For the President had dealt with the matter humbly, in what he conceived as the democratic way.

'', and `` Too Marvelous For Words '' ( all with Richard Whiting ) ; ;

William Wimsatt and Cleanth Brooks, it seems to me, have a penetrating insight into the way in which this control is effected: `` For if we say poetry is to talk of beauty and love ( and yet not aim at exciting erotic emotion or even an emotion of Platonic esteem ) and if it is to talk of anger and murder ( and yet not aim at arousing anger and indignation ) -- then it may be that the poetic way of dealing with these emotions will not be any kind of intensification, compounding, or magnification, or any direct assault upon the affections at all.

For some time, despondency in some Northern quarters had been displayed in two ways -- an eagerness for peace and a dissatisfaction with Lincoln.

For if Serenissimus made the sign of the Cross with his right hand, and meant it, with his left he beckoned lewdly to any lady who happened to catch his eye.

For those little men with the short whiskers, shaven polls, and top knots Suvorov reserved a special esteem.

For some reason, none of them were impressed with the territorial capital.

For example, he captured some persons from York County, who with teams were taking to Philadelphia the furniture of a man who had just been released from prison through the efforts of his wife, and who apparently was helpless to prevent the theft of his household goods.

For as his companions gradually dissolve back into a state of primitive confrontation with elemental necessity, as they lose all the appanage of their acquired culture, he is overcome by the feeling that he is at last being confronted with the essence of mankind.

`` For six years my father had had to do too much commanding and convincing, '' writes the narrator, `` not to understand that man begins with ' the other ' ''.

For a dawning sense of illumination occurs in consequence of two events which, as so often in Malraux, suddenly confront a character with the existential question of the nature and value of human life.

In his recent book, Hurray For Anything ( 1957 ), one of the most important short poems -- and it is the title poem for one of the long jazz arrangements -- is written for recital with jazz.

For a particularly fabulous room which houses a collection of fine English Chippendale furniture, fabric wall panels were embroidered with a typically Chinese-inspired design of this revered Eighteenth Century period.

For the first time in history the entire world is dominated by two large, powerful nations armed with murderous nuclear weapons that make conventional warfare of the past a nullity.

For the case of an object that is small compared with the radial distance to its axis of rotation, such as a tin can swinging from a long string or a planet orbiting in a circle around the Sun, the angular momentum can be expressed as its linear momentum,, crossed by its position from the origin, r. Thus, the angular momentum L of a particle with respect to some point of origin is

For example, in the Schrödinger picture, there is a linear operator U with the property that if an electron is in state right now, then in one minute it will be in the state, the same U for every possible.

For example, observable physical quantities are represented by self-adjoint operators, such as energy or momentum, whereas transformative processes are represented by unitary linear operators such as rotation or the progression of time.

For example, the condition number associated with the linear equation

For simplicity, the following descriptions focus on continuous-time and discrete-time linear systems.

For any n, the divergence is a linear operator, and it satisfies the " product rule "

For the statistics usage, see hierarchical linear modeling.

For other ways to solve this kind of equations, see below, System of linear equations.

For example, according to ancient Hebrew belief, life takes a linear ( and not cyclical ) path ; the world began with God and is constantly headed toward God ’ s final goal for creation.

For an overview of a linear implementation of this framework, see linear regression.

For other examples of quotient objects, see quotient ring, quotient space ( linear algebra ), quotient space ( topology ), and quotient set.

For molecules with N atoms in them, linear molecules have 3N – 5 degrees of vibrational modes, whereas nonlinear molecules have 3N – 6 degrees of vibrational modes ( also called vibrational degrees of freedom ).

For every finite dimensional matrix Lie algebra, there is a linear group ( matrix Lie group ) with this algebra as its Lie algebra.

For a population of size n, with a sequence of values y < sub > i </ sub >, i = 1 to n, that are indexed in non-decreasing order ( y < sub > i </ sub > ≤ y < sub > i + 1 </ sub >), the Lorenz curve is the continuous piecewise linear function connecting the points ( F < sub > i </ sub >, L < sub > i </ sub > ), i

For linear relations, regression analyses here are based on forms of the general linear model.

For example, naive Bayes and linear discriminant analysis are joint probability models, whereas logistic regression is a conditional probability model.

For example, consider the case where the function is a linear function of the form

For example, bacterial chromosomes, plasmids, mitochondrial DNA and chloroplast DNA are usually circular double-stranded DNA molecules, while chromosomes of the eukaryotic nucleus are usually linear double-stranded DNA molecules.

For example, in classical mechanics, the derivative is used ubiquitously, and in quantum mechanics, observables are represented by linear operators.