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Page "Affine transformation" ¶ 15
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Given and two
Euclid poses the problem: " Given two numbers not prime to one another, to find their greatest common measure ".
Given the similarities in the two characters ' names, professions, written works and generally dark subject matter, it is likely that Lovecraft's Alhazred provided the main inspiration for al-Hazir.
Given two subspaces with, this leads to a definition of angles called canonical or principal angles between subspaces.
Given points P < sub > 0 </ sub > and P < sub > 1 </ sub >, a linear Bézier curve is simply a straight line between those two points.
Given the opportunity, adults eat hatchlings, which may be protected by separating the two groups with a net, breeding box or separate tank.
Given that Bozizé accuses Sudan of supporting the UFDR rebels who are actively fighting the Central African Government, the relation between the two countries has remained good.
Given more time to prepare for trial, Kidd likely would have been able to find the deposition Palmer gave when he was captured in Rhode Island two years earlier.
Given two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both
Given that the vast majority of all emeralds are treated as described above, and the fact that two stones that appear visually similar may actually be quite far apart in treatment level and therefore in value, a consumer considering a purchase of an expensive emerald is well advised to insist upon a treatment report from a reputable gemological laboratory.
Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).
Given two groups (< var > G </ var >, *) and (< var > H </ var >, ), a group isomorphism from (< var > G </ var >, *) to (< var > H </ var >, ) is a bijective group homomorphism from < var > G </ var > to < var > H </ var >.
: Given two points, determine the azimuth and length of the line ( straight line, arc or geodesic ) that connects them.
Given these two assumptions, the coordinates of the same event ( a point in space and time ) described in two inertial reference frames are related by a Galilean transformation.
Given two Lie algebras and, their direct sum is the Lie algebra consisting of the vector space
Given this understanding, the Board withdrew the allegation of unsportsmanlike behaviour two days before the fourth Test, thus saving the tour.
Given two secondary stations, the time difference ( TD ) between the primary and first secondary identifies one curve, and the time difference between the primary and second secondary identifies another curve, the intersections of which will determine a geographic point in relation to the position of the three stations.
Given the above commonalities there appear to be only two string theories: the heterotic string theory ( which is also the type I string theory ) and the type II theory.
Given the notion of a lexeme, it is possible to distinguish two kinds of morphological rules.
Given printing practices at the time ( which included type-setting from dictation ), no two editions turned out to be identical, and it is relatively rare to find even two copies that are exactly the same.
Given barely two months to assemble a large sea-going invasion fleet, the Kriegsmarine opted to convert inland river barges into makeshift landing craft.
In the account books of Johanna, Duchess of Brabant and Wenceslaus I, Duke of Luxemburg, an entry dated May 14, 1379 reads: " Given to Monsieur and Madame four peters, two forms, value eight and a half moutons, wherewith to buy a pack of cards ".
# Given any two distinct points, there is exactly one line incident with both of them.
# Given any two distinct lines, there is exactly one point incident with both of them.

Given and affine
Given the affine group of an affine space A, the stabilizer of a point p is isomorphic to the general linear group of the same dimension ( so the stabilizer of a point in Aff ( 2, R ) is isomorphic to GL ( 2, R )); formally, it is the general linear group of the vector space: recall that if one fixes a point, an affine space becomes a vector space.
Given a homogeneous prime ideal P of, let X be a subset of P < sup > n </ sup >( k ) consisting of all roots of polynomials in P .< ref > The definition makes sense since if and only if for any nonzero λ in k .</ ref > Here we show X admits a structure of variety by showing locally it is an affine variety.
Given vector fields and lying in the tangent bundle, the affine connection describes how to differentiate the vector field along the direction.

Given and spaces
Given metric spaces ( X, d < sub > 1 </ sub >) and ( Y, d < sub > 2 </ sub >), a function f: X → Y is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for every x, y ∈ X with d < sub > 1 </ sub >( x, y ) < δ, we have that d < sub > 2 </ sub >( f ( x ), f ( y )) < ε.
Given two metric spaces ( X, d < sub > X </ sub >) and ( Y, d < sub > Y </ sub >), where d < sub > X </ sub > denotes the metric on the set X and d < sub > Y </ sub > is the metric on set Y ( for example, Y might be the set of real numbers R with the metric d < sub > Y </ sub >( x, y )
Given two spaces X and Y, we say they are homotopy equivalent or of the same homotopy type if there exist continuous maps and such that is homotopic to the identity map id < sub > X </ sub > and is homotopic to id < sub > Y </ sub >.
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:
* Given a positive real number ε, an ε-isometry or almost isometry ( also called a Hausdorff approximation ) is a map between metric spaces such that
Given a collection of spaces and maps with maps ( compatible with the inclusions, an X-structure is a lift of ν to a map.
Given a sequence ( X < sub > n </ sub >, p < sub > n </ sub >) of locally compact complete length metric spaces with distinguished points, it converges to ( Y, p ) if for any R > 0 the closed R-balls around p < sub > n </ sub > in X < sub > n </ sub > converge to the closed R-ball around p in Y in the usual Gromov – Hausdorff sense.
Given a pair of spaces, for simplicity we denote.
Given a manifold Q, a vector field X on Q ( or equivalently, a section of the tangent bundle TQ ) can be thought of as a function acting on the cotangent bundle, by the duality between the tangent and cotangent spaces.
Given a pair of spaces ( X, A ) the mapping class group of the pair is the isotopy-classes of automorphisms of the pair, where an automorphism of ( X, A ) is defined as an automorphism of X that preserves A, i. e. f: X → X is invertible and f ( A )
Given two I-graded vector spaces V and W, their direct sum has underlying vector space V ⊕ W with gradation
Given a set X and an indexed family ( Y < sub > i </ sub >)< sub > i ∈ I </ sub > of topological spaces with functions
Given a continuous map there is a map defined by This makes into a functor from the category of topological spaces into itself.
Given a set and a family of topological spaces with functions
Given Borel equivalence relations E and F on Polish spaces X and Y respectively, one says that E is Borel reducible to F, in symbols E ≤< sub > B </ sub > F, if and only if there is a Borel function
Given any desired positive integer c, this theorem shows that one can find an algebraic solution approximating a formal power series solution up to the degree specified by c. This leads to theorems that deduce the existence of certain formal moduli spaces of deformations as schemes.
Given three Hilbert spaces,,

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