When viewed in this way the numbers x < sub > 1 </ sub >, ..., x < sub > n </ sub >, and y < sub > 1 </ sub >, ..., y < sub > n </ sub > are the components of x and y with respect to an orthonormal basis of V.

When the basis is a countable set and orthonormal, that is

When the dimension of H is infinite, then S < sub > 1 </ sub >( H ) ( the trace class ) is not reflexive, because it contains a subspace isomorphic to ℓ < sup > 1 </ sup >, and S < sub >∞</ sub >( H ) = L ( H ) ( the bounded operators ) is not reflexive, because it contains a subspace isomorphic to ℓ < sup >∞</ sup > ( in both cases, the subspace can be chosen to be the operators diagonal with respect to a given orthonormal basis of H ).

When we choose the basis orthonormal: then.

`` When and if it can do so without jeopardizing constitutional and statutory tax-exemption privileges essential to the maintenance of its educational program and facilities, Emory University will consider applications of persons desiring to study or work at the University without regard to race, color or creed, continuing university policy that all applications shall be considered on the basis of intellectual and moral standards and other criteria designed to assure the orderly and effective conduct of the university and the fulfillment of its mission as an institution of Christian higher education ''.

" When Graham arrived in Greenville, Jones, Jr. emphasized that the basis of the university's position was scriptural and not personal.

When the Soviet Union broke up in late 1991, a major boost to Cuba's economy was lost, leaving it essentially paralyzed because of the economy's narrow basis, focused on just a few products with just a few buyers.

When Microsoft later set out to develop a gaming console, the X was used as the basis of the name Xbox to indicate that the console was based on DirectX technology.

When receiving diplomats — who formally represent the sovereign — the receiving head of state grants certain privileges and immunities to ensure they may effectively carry out their duties, on the understanding that these are provided on a reciprocal basis.

When a union of the evangelicals in upper and lower Germany was contemplated as a means of improved defense against the retaliatory measures of the Roman Catholic Church, George had a meeting with Elector John of Saxony at Schleitz in 1529, where they agreed on certain articles of faith and confession to be drawn up by Luther ; the commission was executed in the seventeen articles of Schwabach on the basis of the fifteen theses of the Marburg Colloquy.

When Rousseau subsequently became celebrated as a theorist of education and child-rearing, his abandonment of his children was used by his critics, including Voltaire and Edmund Burke, as the basis for ad hominem attacks.

When Hummay had assumed power on the basis of his strong Islamic following, for example, it is believed that the Duguwa / Zaghawa began some kind of internal opposition.

When obliged to attempt this himself on the basis of the published tables of the day, he always made numerous errors, and never adjusted the figures for his clients ' place or time of birth.

When subjects consumed n-3 from oily fish on a regular basis, their systolic blood pressure was lowered by about 3. 5-5. 5 mmHg.

When Rudolph III, King of Burgundy died on 2 February 1032, Conrad II successfully claimed also this Kingship on the basis of an inheritance Emperor Henry II had extorted from the former in 1006, after having invaded Burgundy to enforce his claim after Rudolph attempted to renounce it in 1016.

When a plea agreement has been made, the judge's factual basis inquiry is usually perfunctory, and the standard for finding that the plea is factually based is very low.

When we measure this qubit in the standard basis, the probability of outcome is and the probability of outcome is.

When Rudolph III, King of Burgundy died 1032, Conrad II also claimed this kingship on the basis of an inheritance Henry II had extorted from the former in 1006.

When the Supreme Court revisited some of the territory covered by Luther v. Borden in cases like Baker v. Carr,, the Fourteenth Amendment's equal protection clause was the basis of its changed decisions.

When distributions are made the taxable portion of the distribution will be calculated as the ratio of the non-Roth contributions to the total 401 ( k ) basis.

To see this, the three corresponding normal vectors of the corner's perpendicular sides can be considered to form a basis ( a rectangular coordinate system ) ( x, y, z ) in which to represent the direction of an arbitrary incoming ray, b, c. When the ray reflects from the first side, say x, the ray's x component, a, is reversed to-a while the y and z components are unchanged, resulting in a direction of b, c. Similarly, when reflected from side y and finally from side z, the b and c components are reversed.

When Rudolph III, King of Burgundy died on 2 February 1032, Conrad claimed the Kingship on the basis of an inheritance Emperor Henry II extorted from the former in 1006, after having invaded Burgundy to enforce his claim after Rudolph attempted to renounce it in 1016.

When cumulus fractus and cumulus humilis are classified as vertical, it is on the basis of their potential for at least moderate upward growth during their daily cycle.

When those plans did not come to fruition, Fleming adapted the ideas to form the basis of the novel, which he originally titled The Wound Man.

When it was revived by Tartaglia and others in the 16th century, it became the basis for quantitative work in science for a century, until it was replaced by the algebraic methods of Descartes.

When the pound was returned to the gold standard after the war it was done on the basis of the pre-war gold price, which, since it was higher than equivalent price in gold, required prices to fall to realign with the higher target value of the pound.

When a guanxi network violates bureaucratic norms, it can lead to corruption, and guanxi can also form the basis of patron – client relations.

When the Soviet collapse and China's marketization switched trade to a hard currency, full price basis, North Korea's economy collapsed.

When it was decided that Asteroids would be a vector game as well, Delman modified a Lunar Lander circuit board for Ed Logg.

* 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 ).

When the angular velocity of this co-rotating frame is not constant, that is, for non-circular orbits, other fictitious forces — the Coriolis force and the Euler force — will arise, but can be ignored since they will cancel each other, yielding a net zero acceleration transverse to the moving radial vector, as required by the starting assumption that the vector co-rotates with the planet.

When applied to vector spaces of functions ( which typically are infinite-dimensional ), dual spaces are employed for defining and studying concepts like measures, distributions, and Hilbert spaces.

When defined for a topological vector space there is a subspace of this dual space, corresponding to continuous linear functionals, which constitutes a continuous dual space.

When the evolution map Φ < sup > t </ sup > ( or the vector field it is derived from ) depends on a parameter μ, the structure of the phase space will also depend on this parameter.

When light crosses an optical element the resulting polarization of the emerging light is found by taking the product of the Jones matrix of the optical element and the Jones vector of the incident light.

When making a gun-like configuration, with the middle finger crossing under the index finger, the fingers represent the velocity vector, magnetic field vector, and force vector, respectively.

When only the magnitude and direction of the vector matter, then the particular initial point is of no importance, and the vector is called a free vector.

When the length of vectors is defined, it is possible to also define a dot product — a scalar-valued product of two vectors — which gives a convenient algebraic characterization of both length ( the square root of the dot product of a vector by itself ) and angle ( a function of the dot product between any two vectors ).

When the coordinates are transformed, for example by rotation or stretching, then the components of the vector also transform.

When considered a vector, weight is often denoted by a bold letter W.

When the system S is prepared in a pure state, it is associated with a vector in H. Measurable quantities are associated with Hermitian operators acting on H: these are referred to as observables.

When new SIMD architectures need to be distinguished from older ones, the newer architectures are then considered " short-vector " architectures, as earlier SIMD and vector supercomputers had vector lengths from 64 to 64, 000.

When applied to a tangent vector X to the surface, the shape operator is the tangential component of the rate of change of the normal vector when moved along a curve on the surface tangent to X.

When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus.