Given two subspaces with, this leads to a definition of angles called canonical or principal angles between subspaces.

Given a vector space V over a field K, the span of a set S ( not necessarily finite ) is defined to be the intersection W of all subspaces of V which contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W.

Given the large number of U. S. military personnel and their dependents residing in Europe, it was expected that over 7 % of donors would be deferred due to the policy.

Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).

Given concerns about the previous programs using nuclear-tipped interceptors, in the 1980s the U. S. Army began studies about the feasibility of hit-to-kill vehicles, i. e. interceptor missiles that would destroy incoming ballistic missiles just by colliding with them head-on.

Given a functor U and an object X as above, there may or may not exist an initial morphism from X to U. If, however, an initial morphism ( A, φ ) does exist then it is essentially unique.

Given current trends in the UMC — with overseas churches growing, especially in Africa, and U. S. churches collectively losing about 1, 000 members a week — it has been estimated that Africans will make up at least 30 % of the delegates at the 2012 General Conference, and it is also possible that 40 % of the delegates will be from outside the U. S. One Congolese bishop has estimated that typical Sunday attendance of the UMC is higher in his country than in the entire United States.

Given control by the U. N. to Magneto until the E Is for Extinction story.

Given Rickover's single-minded focus on naval nuclear propulsion, design and operations, it came as a surprise to many when in 1982, near the end of his career, he testified before the U. S. Congress that, were it up to him, he " would sink them all.

A conceptually very simple method for generating exponential variates is based on inverse transform sampling: Given a random variate U drawn from the uniform distribution on the unit interval ( 0, 1 ), the variate

Given that the U. S. is still home to the world's largest economy, there's no reason it shouldn't have the most vibrant equity markets — unless regulation is holding back the creation of new public companies.

Given Binet's stance that intelligence testing was subject to variability and was not generalizable, it is important to look at the metamorphosis that mental testing took on as it made its way to the U. S.

The American Library Association issued a recommendation to libraries still holding Alms for Jihad: " Given the intense interest in the book, and the desire of readers to learn about the controversy first hand, we recommend that U. S. libraries keep the book available for their users.

Kendall West is home to a large concentration of Colombians and Colombian Americans in the U. S. Given the local name " Little Bogota ".

* Given two complex vectors x and y, multiplication by U preserves their inner product ; that is,

* Given that Historic U. S. Route 66 runs through Elk City, a sprawling museum complex has developed, which includes the National Route 66 Museum, the Old Town Museum, the Transportation Museum, the Farm and Ranch Museum, and the Blacksmith Museum.

Given the opposition to the mission when it began, the primary U. S. concern was to do its limited job and avoid casualties among its forces.

Given the importance of the seasonal gift-giving to the U. S. economy, one driven largely by consumer spending, and with the music industry making at least 40 percent of its annual revenue in the fourth quarter culminating at Christmas, demands for increased revenues motivates the shift.

Given a concrete category ( C, U ) and a cardinal number N, let U < sup > N </ sup > be the functor C → Set determined by U < sup > N </ sup >( c ) = ( U ( c ))< sup > N </ sup >.

Given a Hermitian form Ψ on a complex vector space V, the unitary group U ( Ψ ) is the group of transforms that preserve the form: the transform M such that Ψ ( Mv, Mw ) = Ψ ( v, w ) for all v, w ∈ V. In terms of matrices, representing the form by a matrix denoted, this says that.

# Given u and v in W, then they can be expressed as u = ( u < sub > 1 </ sub >, u < sub > 2 </ sub >, 0 ) and v = ( v < sub > 1 </ sub >, v < sub > 2 </ sub >, 0 ).

# Given u in W and a scalar c in R, if u = ( u < sub > 1 </ sub >, u < sub > 2 </ sub >, 0 ) again, then cu = ( cu < sub > 1 </ sub >, cu < sub > 2 </ sub >, c0 ) = ( cu < sub > 1 </ sub >, cu < sub > 2 </ sub >, 0 ).

Given two C < sup > k </ sup >- vector fields V, W defined on S and a real valued C < sup > k </ sup >- function f defined on S, the two operations scalar multiplication and vector addition

Given the orientation matrix A ( t ) of a frame, we can obtain its instant angular velocity tensor W as follows.

Given a vector x ∈ V and y * ∈ W *, then the tensor product y * ⊗ x corresponds to the map A: W → V given by

Given two normed vector spaces V and W ( over the same base field, either the real numbers R or the complex numbers C ), a linear map A: V → W is continuous if and only if there exists a real number c such that

Given two normed vector spaces V and W, a linear isometry is a linear map f: V → W that preserves the norms:

The Alaska Time Zone applies to the territory of the state of Alaska east of 169 ° 30 ′ W. Given that the UTC − 9 time corresponds to the solar time at 9 × 15 ° = 135 ° W ( roughly, Juneau ), the westernmost locales of the Alaska Time Zone are off by up to 169 ° 30 ′ − 135 ° = 34 ° 30 ′ from local solar time.

Given a relation of order N, consider a projection that subtends X, Y, and Z as the key and W as the residual attribute.

* February 14, 2001: Three Secret Service agents detained and questioned managing editor Glenn Given over a satirical editorial he wrote in which he asked God to " smite " President George W. Bush .< sup ></ sup >

Given the opportunity for accelerating services provided by the opening of the Greenisland Loop Lines and the availability of the new powerful Class W 2-6-0 locomotives, a new express service known as the " North Atlantic Express " was introduced between Portrush on the north coast of County Antrim and Belfast.

Given two I-graded vector spaces V and W, their direct sum has underlying vector space V ⊕ W with gradation

The productions have been: The Invention of Love ( Tom Stoppard, directed by Mary-Kay Gamel, produced by Judith Hallett ), The Heavensgate Deposition ( based on Apocolocyntosis by Seneca the Younger, adapted by Douglass Parker, directed by Amy R. Cohen, produced by Thomas Jenkins ), The Golden Age ( by Thomas Heywood, directed by C. W. Marshall ), Iran Man ( based on Persa by Plautus, directed by Mary-Kay Gamel ), Thespis ( by W. S. Gilbert and A. S. Sullivan, with new music by Alan Riley Jones, directed by John Starks, produced by John Given ), The Birds ( by Aristophanes, directed by Thomas Talboy ), Cyclops ( by Euripides, directed by Laura Lippman and Mike Lippman ), Thersites ( perhaps by Nicholas Udall, directed by C. W. Marshall ), Thesmophoriazusae ( by Aristophanes, directed by Bella Vivante ) and The Jurymen ( by Katherine Janson, directed by Amy R. Cohen ).

Given any vector space V over a field F, the dual space V * is defined as the set of all linear maps ( linear functionals ).

) Given a smooth Φ < sup > t </ sup >, an autonomous vector field can be derived from it.

Given a vector space V over the field R of real numbers, a function is called sublinear if

Given two Lie algebras and, their direct sum is the Lie algebra consisting of the vector space

Given a basis of a vector space, every element of the vector space can be expressed uniquely as a finite linear combination of basis vectors.

Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra can be defined as follows.

Given a vector v in R < sup > n </ sup > one defines the directional derivative of a smooth map ƒ: R < sup > n </ sup >→ R at a point x by

Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:

Given that this is a plane wave, each vector represents the magnitude and direction of the electric field for an entire plane that is perpendicular to the axis.

Given a normalized light vector l ( pointing from the light source toward the surface ) and a normalized plane normal vector n, one can work out the normalized reflected and refracted rays:

Given the dimensions of the ellipsoid, the conversion from lat / lon / height-above-ellipsoid coordinates to X-Y-Z is straightforward — calculate the X-Y-Z for the given lat-lon on the surface of the ellipsoid and add the X-Y-Z vector that is perpendicular to the ellipsoid there and has length equal to the point's height above the ellipsoid.

Given a subset S in R < sup > n </ sup >, a vector field is represented by a vector-valued function V: S → R < sup > n </ sup > in standard Cartesian coordinates ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >).

Given a differentiable manifold M, a vector field on M is an assignment of a tangent vector to each point in M. More precisely, a vector field F is a mapping from M into the tangent bundle TM so that is the identity mapping

Given ω = ( xθ, yθ, zθ ), with v = ( x, y, z ) a unit vector, the correct skew-symmetric matrix form of ω is

Given two column vectors, their dot product can also be obtained by multiplying the transpose of one vector with the other vector and extracting the unique coefficient of the resulting 1 × 1 matrix.