In practical situations there will be restrictions on the admissible operating conditions, and we regard the vectors as belonging to a fixed and bounded set S.

The vector A can be written using any set of basis vectors and corresponding coordinate system.

The great utility in creating CPUs that deal with vectors of data lies in optimizing tasks that tend to require the same operation ( for example, a sum or a dot product ) to be performed on a large set of data.

Start with a set of vectors that are mutually orthogonal.

form an orthogonal basis over the set of N-dimensional complex vectors:

Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a " coordinate system " ( as long as the basis is given a definite order ).

Equivalently, the Fourier transform of such a quasicrystal is nonzero only at a dense set of points spanned by integer multiples of a finite set of basis vectors ( the projections of the primitive reciprocal lattice vectors of the higher-dimensional lattice ).

So, is a set of linearly independent vectors in the column space of and, hence, the dimension of the column space of ( i. e. the column rank of ) must be at least as big as.

It is a generalization of the statement that if the span of n vectors has dimension p, then p of those vectors span the space: one can choose a spanning set that is a subset of the vectors.

It can be alternatively stated as: if the span of n vectors has dimension p, then p of these vectors span the space and there is a set of p coordinates on which they are linearly independent.

Similarly the set of all vectors in which is rational for all i is a countable dense subset of ; so for every the-dimensional Euclidean space is separable.

* Euclidean subspace, in linear algebra, a set of vectors in n-dimensional Euclidean space that is closed under addition and scalar multiplication

The tangent space of M at x, denoted by T < sub > x </ sub > M, is defined as the set of all tangent vectors ; it does not depend on the choice of chart φ.

Typically, these components are the projections of the vector on a set of reference axes ( or basis vectors ).

A matrix M is positive-semidefinite if and only if it arises as the Gram matrix of some set of vectors.

It works by dividing a large set of points ( vectors ) into groups having approximately the same number of points closest to them.

It is compressed by choosing the nearest matching vector from a set of N-dimensional vectors.

It appears possible to set a lower limit of about Af for the activation energy of the abstraction of a chlorine atom from a carbon tetrachloride molecule by a chlorine atom to form Af radical.

When each number of successes X is paired with its probability of occurrence Af, the set of pairs Af, is a probability function called a binomial distribution.

This theorem is similar to the theorem of Kakutani that there exists a circumscribing cube around any closed, bounded convex set in Af.

The set of intersections of Af, the rotated curve, with the original curve C consists of just the set of forward corner points on C corresponding to the vertex at Af, plus the vertex itself.

In the C-plane we construct a set of rectangular Cartesian coordinates u, V with the origin at Q and such that both C and Af have finite slope at Q.

The fact that there can not be any limit points of the set except in closed intervals follows from the argument used in Lemma 1, namely, that near any tangent point in the C-plane the curves C and Af are analytic, and therefore the difference between them must be a monotone function in some neighborhood on either side of the tangent point.

I was at least conscious of the distinction in my full Yokuts presentation that awaits publication, in which, in listing ' Two-Stem Meanings ', I set off by asterisks those forms in which N of stem B was Af of stem A/3, the unasterisked ones standing for Af ; ;

or under ' Four Stems ', I set off by asterisks cases where the combined N of stems Af was Af.

In the Sermon on the Mount he set forth a new commandment concerning marriage, teaching that lustful looking constitutes adultery.

As the preface says, " Not all scholars share the same set of criteria for what constitutes a ' language ' and what features define a ' dialect.

Because no set definition of what constitutes a lake and no procedures for counting the number of lakes exist, it has been impossible to ascertain exactly how many lakes the region has.

" We'd be oversimplifying things in calling film noir oneiric, strange, erotic, ambivalent, and cruel ": this set of attributes constitutes the first of many attempts to define film noir made by French critics Raymond Borde and Etienne Chaumeton in their 1955 book Panorama du film noir américain 1941 – 1953 ( A Panorama of American Film Noir ), the original and seminal extended treatment of the subject.

An answer is a pleading filed by a defendant which admits or denies the specific allegations set forth in a complaint and constitutes a general appearance by a defendant.

In essence, a system constitutes a set of interrelated components working together with a common objective: fulfilling some designated need.

This set constitutes some of the earliest heptasyllabic shi Chinese poetry written.

Chanchal Kumar Sharma ( 2005a, 2005b ) clarifies: While fiscal federalism constitutes a set of guiding principles, a guiding concept, that helps in designing financial relations between the national and subnational levels of the government, fiscal decentralization on the other hand is a process of applying such principles ( Sharma, 2005b: 178 ).

The set of all possible spin networks ( or, more accurately, " s-knots "-that is, equivalence classes of spin networks under diffeomorphisms ) is countable ; it constitutes a basis of LQG Hilbert space.

The Nuremberg principles were a set of guidelines for determining what constitutes a war crime.

It lists a set of disorders and provides detailed descriptions on what constitutes a disorder such as Major Depressive Disorder or anxiety disorder.

Haggetts Pond was originally set apart from other waters, but since the late 1990s has had waters added from the Merrimack River, which constitutes half of the town's northern border, to supplement the growing needs of the town.

Spark imbues Caroline, her central character, with voices in her head which constitutes the narration Spark has just set down on the page.

The AB-AC list was used because its second set of associations ( A-C ) constitutes a modification of the first set of associations ( A-B ), whereas the AB-DE list were used as a control.

Chanchal Kumar Sharma clarifies that while fiscal federalism constitutes a set of guiding principles, a guiding concept that helps in designing financial relations between the national and subnational levels of the government, fiscal decentralization on the other hand is a process of applying such principles.

A set of folds distributed on a regional scale constitutes a fold belt, a common feature of orogenic zones.

In fact, the collection of all such convex combinations of points in the set constitutes the convex hull of the set.

As the level of difficulty increases, individuals increasingly demonstrate an understanding of what constitutes a key difference ( or the " rule ") for solving puzzles involving one to one comparisons, and on later items identifying common differences among a set of items.

Issues of different national customs can arise – while the U. S. and host countries generally agree on what constitutes a crime, many U. S. observers feel that host country justice systems grant a much weaker set of protections to the accused than the U. S. and that the host country's courts can be subject to popular pressure to deliver a guilty verdict ; furthermore, that American servicemembers ordered to a foreign posting should not be forced to give up the rights they are afforded under the Bill of Rights.

For example, it is not irrational to set a trap for someone who is wearing an Armani suit so that they fall into a swimming pool in front of a video camera, since the pleasure one can get out of watching the video constitutes an adequate reason for harming the other person.

Clondalkin may be set to become one of Ireland's newest Gaeltacht regions thanks to a Government proposal to redefine what constitutes a Gaeltacht, " based on linguistic criteria instead of on geographic areas ".

An honor code or honor system is a set of rules or ethical principles governing a community based on ideals that define what constitutes honorable behavior within that community.