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Topic: Affine subspace

	Affine transformation - Wikipedia, the free encyclopedia
		An affine transformation is invertible iff A is invertible.
		The invertible affine transformation form the affine group, which has the general linear group of degree n as subgroup and itself is a subgroup of the general linear group of degree n+1.
		An affine subspace of a vector space (sometimes called a linear manifold) is a coset of a linear subspace; i.e., it is the result of adding a constant vector to every element of the linear subspace.
	en.wikipedia.org /wiki/Affine_transformation (1058 words)

	Encyclopedia: Affine transformation
		In geometry, an affine transformation or affine map (from the Latin, affinis, "connected with") between two vector spaces consists of a linear transformation followed by a translation.
		In mathematics, an affine combination of vectors x1,..., xn is a linear combination in which the sum of the coefficients is 1, thus:.
		The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics.
	www.nationmaster.com /encyclopedia/Affine-transformation (1167 words)

	info/guide/a/af/affine_combination - Info and Guide. (Site not responding. Last check: 2007-10-15)
		Affine space - Affine space In mathematics, an affine space may be defined somewhat abstractly as a set on which a vector space acts transitively.
		Affine transformation - Affine transformation An affine transformation or affine map (from the Latin, affinis, "connected with") between two vector spaces consists of a linear transformation followed by a translation.
		A linear subspace of a vector space is a subset that is closed under linear combinations; an.
	pheeds.com /info/guide/a/af/affine_combination.html (2248 words)

	Simplex - Wikipedia, the free encyclopedia
		Such a general simplex is often called an affine n-simplex, to emphasize that the canonical map is an affine transformation.
		It is also sometimes called an oriented affine n-simplex to emphasize that the canonical map may be orientation preserving or reversing.
		Note that each face of an n-simplex is an affine n-1-simplex, and thus the boundary of an n-simplex is an affine n-1-chain.
	en.wikipedia.org /wiki/Simplex (833 words)

	Affine map (Site not responding. Last check: 2007-10-15)
		An affine transformation or affine map (from the Latin, affinis, "connected with")between two vector spaces consists of a linear transformation followed by a translation.
		An affine subspace of a vector space is a coset of a linear subspace ; i.e., it is the result of adding a constant vector to everyelement of the linear subspace.
		A linear subspace of a vector space is a subset that is closed under linear combinations; anaffine subspace is one that is closed under affine combinations.
	www.therfcc.org /affine-map-270237.html (349 words)

	Affine transformation -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-15)
		An affine subspace of a vector space is a (Click link for more info and facts about coset) coset of a (Click link for more info and facts about linear subspace) linear subspace; i.e., it is the result of adding a constant vector to every element of the linear subspace.
		The set of all invertible affine transformations forms a ((chemistry) two or more atoms bound together as a single unit and forming part of a molecule) group under the operation of composition of functions.
		To visualise the general affine transformation of the (Click link for more info and facts about Euclidean plane) Euclidean plane, take labelled (A quadrilateral whose opposite sides are both parallel and equal in length) parallelograms ABCD and A′B′C′D′.
	www.absoluteastronomy.com /encyclopedia/a/af/affine_transformation.htm (734 words)

	Introduction
		Affine transformations are transformations that preserve collinearity of points.
		Affine mappings are of the form Ax + b where A is an nxn square matrix and x and b are vectors in
		Affine geometry is closely related to projective geometry because the set of affine transformations is a subgroup of the general linear transformations on the projective plane, so affine space is embedded in the projective space
	www.math.umd.edu /~lidador/Affine/home.html (387 words)

	Z4=An Electronic Display
		Classical affine and projective spaces have a simple relationship: removing part of a projective space leaves an affine subspace of the same dimension as the original projective space.
		It appears that desarguesian affine spaces are important for physics because any two points in such a space are connected by a unique vector from a classical vector space.
		As an illustration of a desarguesian affine plane, consider the relatively simple 13 point projective plane with an affine subplane of 9 distinct points.
	www.geocities.com /horst1925/projtext.html (1273 words)

	Dictionary of Meaning www.mauspfeil.net (Site not responding. Last check: 2007-10-15)
		In geometry, an '''affine transformation''' or '''affine map''' (from the Latin, ''affinis'', "connected with") between two vector spaces consists of a linear transformation followed by a translation (geometry) translation.
		The set of all invertible affine transformations forms a group (mathematics) group under the operation of composition of functions.
		Interesting that "Affine transformations of the plane" specifies a parallelogram as the basic unit defining an affine transformation.
	www.mauspfeil.net /Affine_transformation.html (1147 words)

	[No title]
		The original system is span-reachable iff the second one is. The advantage of this normalization is that since.~=O is now always in the affine span of the reachable set, this span is a subspace and span-reachability means X=subspace generated by g(U*).
		Defining L_span reachable to mean that X is the smallest subspace (rather than affine manifold) generated by reachable states and defining L-span canonical: = L-span reachable + observable, L-morphism: = linear T as in (1.7), the proofs in the previous sections can be repeated with minor modifications to yield the following theorem.
		In the proof of algorithm (2.8) let Xj be the subspace of X generated by all those columns B., a in [J],, for whichj or more of the ai are nonzero.
	www.math.rutgers.edu /~sontag/FTP_DIR/state-affine-realiz.txt (8139 words)

	[No title] (Site not responding. Last check: 2007-10-15)
		Affine schemes \endhead \vskip 0.7cm Returning to the commutative setting let $R$ be again a commutative ring with unit $1$.
		Affine schemes are the ``model spaces'' of algebraic geometry.
		So, given a ring $R$ its associated affine scheme is the pair $\ (\Spec(R),\Or)\ $ where $\ \Spec(R)\ $ is the set of prime ideals made into a topological space by the Zariski topology and $\Or$ is a sheaf of rings on $\Spec(R)$ which we will define in a minute.
	www.math.uni-mannheim.de /~schlich/preprints/algeo.tex (12290 words)

	Martina Finzel-Hoffmanns Habilitationsschrift
		They are all piecewise affine and globally Lipschitz continuous, and, in addition, if G is a linear subspace, the strict best approximation is quasi-linear with respect to G.
		Elementary vectors of a subspace are vectors therein which have minimal support.
		If G is a linear subspace the tuples of the classification are exactly the supports of the elementary vectors of the orthogonal complement of G.
	www.mi.uni-erlangen.de /~approx/THESES/finzel.html (621 words)

	Affine transformation
		An affine transformation or affine map between two vector spaces consists of a linear transformation followed by a translation.
		Intuitively, these are precisely the functions that map straight lines to straight lines.
		The text of this article is licensed under the GFDL.
	www.ebroadcast.com.au /lookup/encyclopedia/af/Affine_transformation.html (284 words)

	Affine Geometry (Site not responding. Last check: 2007-10-15)
		Affine transformations map affine subspaces to affine subspaces.
		This follows from the fact that linear maps map linear subspaces to linear subspaces.
		Since affine transformations preserve planes and incidence, their images lie in an affine plane and do not meet.
	www-groups.dcs.st-and.ac.uk /~john/geometry/Lectures/L13.html (229 words)

	Fields Institute - The Coxeter Legacy
		In prime characteristic, the geometry of cells in the affine Weyl group as well as the geometry of the flag variety and the nilpotent variety are conjecturally involved in many of the unsolved problems.
		In the case when the affine subspace is a hyperplane such estimates were found in [1].
		In fact, the case of an $l$-dimensional affine section is completely analogous to the case of a hyperplane section.
	www.fields.utoronto.ca /programs/scientific/03-04/coxeterlegacy/abstracts.html (2690 words)

	Selected Matches for: Items Authored by Clark, W. Edwin
		A hyperplane in $V$ is a translation of an $(n-1)$-dimensional subspace $W$ of $V$ and the set of all translations of $W$ is called a parallel class of hyperplanes.
		An affine semigroup $S$ is a linear variety (i.e., a translate of a subspace of a vector space) endowed with an associative multiplication for which the mappings $x\rightarrow xa$ and $x\rightarrow ax (x\in S)$ are affine mappings for all $a\in S$.
		Affine semigroups are characterized as follows: They are semigroups $S$ isomorphic to semigroups $\phi\sp {-1}(1)$, where $\phi$ is a $\Phi$-epimorphism of an algebra $A$ over $\Phi$ onto the field $\Phi$ and $\phi\sp {-1}(1)$ is regarded as a subsemigroup of the multiplicative semigroup of $A$.
	www.math.usf.edu /~eclark/pubs_mathscinet.html (6661 words)

	[cryst] 2 Affine crystallographic groups
		In the representation by augmented matrices, affine crystallographic groups are infinite matrix groups.
		The connection line is labelled with the number of affine subspaces contained in the lower Wyckoff position that contain a fixed representative affine subspace of the upper Wyckoff position.
		For instance, if the lower Wyckoff position consists of a space group orbit of lines (and thus the upper one of an orbit of points), the label of the connection line is the number of lines in the orbit which cross a fixed representative point of the upper Wyckoff position.
	www.gap-system.org /oldsite/pkg/cryst/htm/CHAP002.htm (2576 words)

	Maps between Schemes
		The restriction of the map f, a map of schemes from an affine scheme to a projective scheme, to a rational map from its domain to the jth standard affine patch of its codomain.
		The linear automorphism of the affine space A determined by the matrix of ring elements M (acting on the left of coordinate functions).
		The automorphism of the affine space A that permutes its coordinates according to the permutation g.
	www.math.niu.edu /help/math/magmahelp/text974.html (5554 words)

	Affine Spaces (Site not responding. Last check: 2007-10-15)
		S is a subspace if it is closed under linear combinations.
		S is an affine space if it is closed under affine combinations.
		An affine space is a translation of a subspace.
	www.rpi.edu /~mitchj/matp6640/affine/affine.html (183 words)

	Patent 5159512
		An affine subspace of R.sup.n is a translation of a linear subspace of R.sup.n.
		An affine cell in R.sup.n is a relatively open (in the affine space topology) connected subset of an affine subspace of R.sup.n.
		The k-dimensional affine space that contains an affine k-cell, c, as a relatively open subset is denoted by V(c), and the k-dimensional space of vectors tangent to c, by T(c).
	www.freepatentsonline.com /5159512.html (14935 words)

	Algorithms for Solving Linear Systems
		These methods are characterized by the subspaces in which the iterates lie.
		Different Krylov subspace methods differ in the criteria they use in selecting a vector in the subspace.
		A number of Krylov subspace methods have been developed for the case when the matrix is not symmetric, positive definite.
	www.ncsa.uiuc.edu /News/datalink/0212/lci/node4.html (563 words)

	degrees of freedom of 2d affine matrix (Site not responding. Last check: 2007-10-15)
		First of all, you seems to be messing affine and linear transforms.
		affine had no translation component, whereas linear did.
		of equations AX = B is an affine* subspace of a finite
	www.groupsrv.com /computers/about87293.html (1985 words)

	An affine transformation or affine map from the Latin... (Site not responding. Last check: 2007-10-15)
		An affine transformation or affine map from the Latin...
		Such a vector ("a"1,..., "a"n") is an "affine dependence" among the vectors "v"1, "v"2,..., "v"n".
		That group is called the affine group, and is the semidirect product of "K"n" and GL("n", "k").
	www.geodatabase.de /affine%20transformation (391 words)

	[No title]
		More precisely, there is a subspace Z of the test space VP such that a configuration C 2 MP is a solution if and only if f(C) 2 Z. The inner symmetries of the problem P typically show up at this stage.
		The "zero" subspace ZP is defined as the trivial, 1-dimensional Wn-representation V0 contained in V.
		Assume as before that R4 is identified to the affine subspace R4+ e5 R5.
	hopf.math.purdue.edu /Zivaljevic/synergia.txt (7026 words)

	Citations: An adaptive filtering algorithm using an orthogonal projection to an affine subspace and its properties - ... (Site not responding. Last check: 2007-10-15)
		An adaptive filtering algorithm using an orthogonal projection to an affine subspace and its properties.
		The algorithm can be described as a regularized version of the recursive least squares algorithm with an N element sliding rectangular window, where N L [6] For a full derivation of the algorithm, the reader is referred to [4, 5, 6] we only summarize the algorithm derivation here.
		In APA, the weight vector update is obtained from a projection on an affine subspace with dimension L Gamma N, where L is the length of the filter and N is an integer.
	citeseer.lcs.mit.edu /context/498028/0 (905 words)

	MA243 - GEOMETRY - Week 4 (Site not responding. Last check: 2007-10-15)
		I introduce the notion of dimension, affine linear subspace, affine linear combinations, and affine linear span, and prove a formula for the dimension of intersections of affine linear subspaces.
		As an application of affine geometry, we can prove that the three medians of a triangle meet in its centroid.
		Note that the concept of affine space can be viewed as first success of Klein's viewpoint over the Euclidean one.
	www.maths.warwick.ac.uk /~wendland/ma243/week3.html (235 words)

	Computer Graphics : Mathematics : 6 / 23 : Various other curiosities
		Equation of a Line in an Affine Space defined by two point P and Q and t a real
		Equation of a Plane in an Affine Space defined by three not collinear point P, Q and R
		Affine Subspace of R4 : standard affine 3 space
	escience.anu.edu.au /lecture/cg/Maths/curiousities.en.html (195 words)

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