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	math lessons - Dual space
		Given any vector space V over some field F, we define the dual space V* to be the set of all linear functionals on F, i.e., scalar-valued linear transformations on V (in this context, a "scalar" is a member of the base-field F).
		f produces an injective homomorphism between the space of linear operators from V to W and the space of linear operators from W* to V*; this homomorphism is an isomorphism iff W is finite-dimensional.
		In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over F to itself.
	www.mathdaily.com /lessons/Dual_space (964 words)

	Dual space - Wikipedia, the free encyclopedia
		produces an injective homomorphism between the space of linear operators from V to W and the space of linear operators from W* to V*; this homomorphism is an isomorphism iff W is finite-dimensional.
		In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over F to itself.
		The continuous dual V′ of a normed vector space V (e.g., a Banach space or a Hilbert space) forms a normed vector space.
	en.wikipedia.org /wiki/Dual_space (1019 words)

	Associated Vector Space (Site not responding. Last check: 2007-10-18)
		return the underlying vector space, dual vector space, and lattice associated to a space of modular symbols.
		A space of modular symbols is represented internally as a subspace of a vector space, and a subspace of the linear dual of the vector space.
		The lattice generated by the integral modular symbols in the vector space representation of M. This is the lattice generated by all modular symbols X^iY^(k - 2 - i){a, b}.
	www.umich.edu /~gpcc/scs/magma/text1103.htm (146 words)

	Dual space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-18)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, the existence of a 'dual' vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1).
		If V consists of the space of geometrical (A variable quantity that can be resolved into components) vectors (arrows) in the plane, then the elements of the dual V* can be intuitively represented as collections of parallel lines.
		Spaces for which the map Ψ is a (Click link for more info and facts about bijection) bijection are called (A personal pronoun compounded with -self to show the agent's action affects the agent) reflexive.
	www.absoluteastronomy.com /encyclopedia/d/du/dual_space.htm (1120 words)

	Encyclopedia: Functional analysis (Site not responding. Last check: 2007-10-18)
		In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear functionals.
		The dual of the dual is not always isomorphic to the original space, but there is always a natural monomorphism from a space into its dual's dual.
		In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...
	www.nationmaster.com /encyclopedia/Functional-analysis (2281 words)

	Vector Spaces.
		If any member of a sub-set of the vector space is in the span of the rest of the sub-set, the sub-set is said to be linearly dependent; conversely, if no member is in the span of the rest, the sub-set is called linearly independent.
		It is possible to show that every basis of a vector space (that has at least one finite basis) has the same number of members: the number in question is known as the dimension of the vector space; a vector space with dimension n is described as n-dimensional.
		Vector spaces are widely used in modelling physical systems since, at least to a high degree of accuracy, much of physics conforms to the character of vector spaces.
	www.chaos.org.uk /~eddy/math/vector.html (1388 words)

	Methods of Mathematical Physics (PHYS 607) at University of Delaware
		Scalar, vector, and inner products.Vector spaces (finite dimensional).
		Linear functional and dual vector space .
		Vectors and matrices with Computer Algebra packages and with LAPACK.
	www.physics.udel.edu /~bnikolic/teaching/phys607/phys607.html (804 words)

	PlanetMath: dual space
		The notions of duality extend, in part, from vector spaces to modules, especially free modules over commutative rings.
		A related notion is the duality in projective spaces.
		This is version 10 of dual space, born on 2002-02-03, modified 2004-09-17.
	planetmath.org /encyclopedia/DualSpace.html (235 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-18)
		I thought that a dual space was when you have a vector space V and you have a transformation L that takes V to another vector space U. And a dual map is a function that assigns U* to a field in V*.
		Any vector parallel or anti-parallel to the z-axis has x=0 and y=0, and thus L maps any vector that points straight up in the +z direction, or straight down in the -z direction, to the zero vector.
		The set of vectors which are orthogonal to all vectors in the x-y plane are all vectors which point straight up in the +z direction or straight down in the -z direction.
	mathforum.org /library/drmath/view/63857.html (1088 words)

	Normed vector space (Site not responding. Last check: 2007-10-18)
		If V is a vector space over a field K (which must be either the real numbers or the complex numbers), a norm on V is a function from V to R, the real numbers — that is, it associates to each vector v in V a real number, which is usually denoted
		A surjective isometry between the normed vector spaces V and W is called a isometric isomorphism, and V and W are called isometrically isomorphic.
		The dual V ' of a normed vector space V is the space of all continuous linear maps from V to the base field (the complexes or the reals) — such linear maps are called "functionals".
	www.city-search.org /no/normed-vector-space.html (1103 words)

	BEND, Future Directions/Improvements, Page 1
		Internally the rulings should be represented by dual numbers because lines in space can be represented by dual vectors whose inner-product-square equals one (see §9.4 in Selig and page 156 in Pottmann and Wallner).
		Now, confusingly, we must use the word "dual" in a different sense: Given any vector space there is automatically given another, the space of linear functionals on it which is, by a permanently entrenched mathematical convention, called its dual.
		The dual of a vector space (in our finite-dimensional case) looks just like the vector space of which it is the dual, and the dual of the dual is the original space so they come in pairs.
	tdserver1.fnal.gov /cook/FutBa.htm (5544 words)

	Top Page 1 (Site not responding. Last check: 2007-10-18)
		In the dual vector field, the left-handed tetrahedron, is isomorphic to the field of the right-handed tetrahedron.
		The V4 vertices of its dual, the rhombic dodecahedron, are the center of the cube's faces, which are determined by the crossing of the perpendicular edges of a chiral pair of dual tetrahedra that are coincident at their midpoints.
		The dodecahedron, the dual of the former, is mapped to the inside surface, with its thirty edges orthogonal to its dual.
	www.mi.sanu.ac.yu /vismath/dyke (6091 words)

	Lp space (Site not responding. Last check: 2007-10-18)
		spaces are spaces of p-power integrable functions, and corresponding sequence spaces.
		They form an important class of examples of Banach spaces in functional analysis, and of topological vector spaces.
		spaces, in which the measure used in the integration in the definition is a counting measure and the measure space S is discrete.
	www.worldhistory.com /wiki/L/Lp-space.htm (648 words)

	Prerequisites
		A vector space over a field F is an additive group (V,+,0) with a scalar multiplication F × V
		v in V} is a vector space, the quotient space of V by W.
		The dual vector space of a vector space V over F is Hom(V,F).
	www.win.tue.nl /~amc/ow/lba/voorkennisvs.html (595 words)

	Read This: Geometrical Vectors
		From this standpoint, physicists talk of "contravariant vectors" (the usual ones), "covariant vectors" (elements of the dual vector space), and of even stranger beasts such as "contravariant vector densities".
		There is clearly a way to translate such a stack vector to an arrow: take the arrow perpendicular to the family of parallel planes, of a length proportional to the density of the stack.
		To the mathematician, this is the clue: "stack vectors" are elements of the vector space dual to the space of arrows.
	www.maa.org /reviews/vectors.html (1359 words)

	C++ page (Site not responding. Last check: 2007-10-18)
		As members of a vector space, vectors may be added and multiplied by scalers.
		It may be shown that the dimension the Grassmann algebra is two to the nth power where n is the dimension of the vector space.
		In physics we have the differential forms which are elements of the Grassmann Algebra over the dual vector space V*.
	www.physics.uci.edu /~rader/project/overview.html (411 words)

	One-forms - Page 2 - Physics Help and Math Help - Physics Forums
		For instance, for a sphere imbedded in Euclidean 3-space, the tangent vector space at a point P on the sphere would be the plane that touches the sphere tangentially at the point P. As the tangent space is a vector space, we can define one-forms on it.
		That dual space is the space of one-forms at P. However, there is a typo in his description.
		Schutz, you, me have no problem in thinking about a vector space Tp at P which is dual* to the vector space Tp at P. That doesn't differ in any way from what I was taught before I'd ever heard of 1-forms.
	www.physicsforums.com /showthread.php?p=809373#post809373 (1595 words)

	Bra space (Site not responding. Last check: 2007-10-18)
		This type of vector space is called a bra space (after Dirac) and its constituent vectors (which are actually functionals of the ket space) are called bra vectors.
		Recall that a bra vector is a functional which acts on a general ket vector and spits out a complex number.
		, and is analogous to the length, or magnitude, of a conventional vector.
	farside.ph.utexas.edu /teaching/qm/fundamental/node9.html (567 words)

	week83
		There is an interesting analogy between the dual of a vector space and the inverse of a number which I would like to explain now.
		Well, the whole point of the dual vector space y is that a vector in y is a linear functional from x to 1.
		So we see that dual vector spaces are a bit like inverse numbers, except that we don't have yx = 1 and 1 = xy, and we don't even have that yx is isomorphic to 1 and 1 is isomorphic to xy.
	math.ucr.edu /home/baez/week83.html (1752 words)

	PlanetMath: dual space
		Linear forms are also known as linear functionals.
		algebraic dual, continuous dual, dual basis, reflexive, natural embedding
		Cross-references: projective spaces, commutative rings, free modules, modules, bilinear form, Riesz representation theorem, Hilbert space, finite dimensional, continuous, subspace, topological vector space, injection, independent, easy to see, mapping, bijection, natural isomorphism, canonical, finite-dimensional, strictly, basis, cardinal, infinite, finite, dimension, isomorphic, operations, linear mappings, linear forms, field, bases, vector space
	planetmath.org /encyclopedia/DualBasis.html (235 words)

	dual from FOLDOC (Site not responding. Last check: 2007-10-18)
		Loosely, where there is some binary symmetry of a theory, the image of what you look at normally under this symmetry is referred to as the dual of your normal things.
		There is a natural embedding of any vector space in the dual of its dual:
		It is conventional, when talking about vectors in V, to refer to the members of V' as covectors.
	www.instantweb.com /d/dictionary/foldoc.cgi?query=DUAL (153 words)

	Dual space : Duality (linear algebra) (Site not responding. Last check: 2007-10-18)
		In mathematics the existence of a 'dual' vector space reflects in an abstract way the relationship between row vectors (1xn) and column vectors (nx1).
		Given any vector space V over some field F, we define the dual space V* to be the set of all linear functions from V to F. These linear functions to the base field are also called linear functionals.
		of all sequences space of sequences a = (a
	www.city-search.org /du/duality-(linear-algebra).html (1294 words)

	CONTROL and DYNAMICAL SYSTEMS - California Insitute of Technology (Site not responding. Last check: 2007-10-18)
		Motivation for geometric methods: general, historic introduction Example of pendulum: introduce manifolds as configuration spaces; differentiation leads to notion of tangent space; vector fields on manifolds as fundamental object.
		Topology: motivation; metric spaces; topological spaces; mappings between topological spaces; properties.
		Vector fields: integral curves; flow of a vector field; coordinate-free definition of dynamical system.
	www.cds.caltech.edu /courses/2002-2003/cds202/coursetext.html (242 words)

	[No title]
		New notation is intro- duced here in order to identify the vector space V explicitly, and to avoid a clash with the notation cr(ae) for Chern classes.
		Take the cor- responding dual basis A1, : :,:Bnfor E, and recall that one-dimensional representations of G are identified with elements of E.
		Bases In this section, we establish a result about the restrictions of the Ajand Bjto the dual space I, for any maximal totally isotropic subspace I of E. To this end, we shall introduce a symplectic form bL on E, which will also play an important role in subsequent sections.
	hopf.math.purdue.edu /Green-Leary/extra.txt (4759 words)

	[No title]
		Via # the group G acts on the vector space V = IF n, and hence also on the graded algebra IF[V ] of homogeneous polynomial functions on V.
		Dual to the map a is the Noether map g : V ## map(G, V) which is a Gequivariant map, and hence, induces a Gequivariant map g * : IF[W] # IF[V ], which we also refer to as the Noether map, though no confusion should arise.
		Let Z be the vector space dual of b Å i=1 A i, where A i denotes the homogeneous component of A in degree i.
	hopf.math.purdue.edu /LSmith/wreath.txt (2804 words)

	Usenet Archive
		I guess then that in the > > perceptual space, a new born baby would be the analogy of a singularity > > (big bang rather than a fl hole).
		In the inter-galactic space, the density of gravitational ether is minimal, with maximums occurring in the centre of fl holes.
		Thus the quantised gravitational field would imply fluctuations in the field and since the gravitational potential reflects the metric properties of the space, the space-time itself must be fluctuating.
	www2.usenetarchive.org /Dir68/File907.html (10953 words)

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