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Topic: Dimension theorem for vector spaces

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	Vector Spaces -- Recommendations and Resources (Site not responding. Last check: 2007-10-13)
		A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra.
		A semi normed vector space is a 2-tuple (''V'',''p'') where ''V'' is a vector space and ''p'' a semi norm on ''V''.
		In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e.
	www.becomingapediatrician.com /health/156/vector-spaces.html (1125 words)

	PlanetMath: theorem for the direct sum of finite dimensional vector spaces
		be subspaces of a finite dimensional vector space
		"theorem for the direct sum of finite dimensional vector spaces" is owned by matte.
		This is version 5 of theorem for the direct sum of finite dimensional vector spaces, born on 2003-04-29, modified 2006-10-14.
	planetmath.org /encyclopedia/ATheoremForTheDirectSumOfFiniteDimensionalVectorSpaces.html (134 words)

	Differential Forms and the Generalized Stokes Theorem (Site not responding. Last check: 2007-10-13)
		Roughly, a vector space is a set of entities such that the sum of any vectors is also a vector and the result of multiplying a vector by a scalar is also a vector.
		Formally the vector space is a set of four things, (V,k,+,) where V is the set of vectors, K is the field of scalars involved in creating multiples of vectors, + is the function involved in adding two vectors and * is the binary function involved in multiplying a vector by a scalar.
		The vector space of linear functionals over V is said to be dual to the vector space involving V. For vector spaces with finite bases the dual spaces are not very exotic; they are essentially the same as the original spaces.
	www.sjsu.edu /faculty/watkins/difforms.htm (1932 words)

	Dimension (vector space) - Wikipedia, the free encyclopedia
		In mathematics, the dimension of a vector space V is the cardinality (i.e.
		All bases of a vector space have equal cardinality (see dimension theorem for vector spaces) and so the dimension of a vector space is uniquely defined.
		One can see a vector space as a particular case of a matroid, and in the latter there is a well defined notion of dimension.
	en.wikipedia.org /wiki/Hamel_dimension (501 words)

	Real Vector Spaces
		Since a vector space has a constant number of vectors in a basis, that number n is characteristic for that space and is called the dimension of that space.
		The space generated by D is called the row space of A. The rows of A are a generating set of the row space.
		A is the supplementary vector space of B with respect to V. B is the supplementary vector space of A with respect to V. A and B are supplementary vector spaces with respect to V. Basis and direct sum
	www.ping.be /~ping1339/vect.htm (4070 words)

	Vector (spatial) Summary
		A spatial vector is a special case of a tensor and is also analogous to a four-vector in relativity (and is sometimes therefore called a three-vector in reference to the three spatial dimensions, although this term also has another meaning for p-vectors of differential geometry).
		Also, let, for example, a vector field be expressed as three space coordinate functions of three variables, and apply the formula for the curl based on these functions, resulting in three additional functions, which represent a second vector field.
		Vectors can be contrasted with scalar quantities such as distance, speed, energy, time, temperature, charge, power, work, and mass, which have magnitude, but no direction (they are invariant under coordinate rotations).
	www.bookrags.com /Vector_(spatial) (5261 words)

	Vector Spaces
		A Vector Space is a set, attributed to a field, that has one internal law of composition and one external law of composition.
		The field that the vector space is taken with respect to is known as the scalar field.
		A vector space is said to have a dimension of n if a basis in it has n elements.
	www.angelfire.com /geek/mathrealm/vector.html (1268 words)

	Geometry Seminar Lecture Notes 1
		Theorem: The automorphism group of GF(q) of characteristic p is cyclic and generated by the Frobenius automorphism.
		A rank 1 subspace of V consists of all the scalar multiples of a given vector, thus there are q vectors in such a subspace (including the zero vector).
		The word "dimension" is used here in the classical geometric sense in which lines have 1 dimension, planes have 2 dimensions, etc. This use of the term is different from (but related to) the algebraic dimension of vector spaces (rank).
	www-math.cudenver.edu /~wcherowi/geom/gsln1.html (1161 words)

	Linear Vector Spaces, and Subspaces
		Theorem: Every basis of a given vector space have the number of basis vectors or dimension.
		N is the dimension of the vector space, we can always construct a basis by adding additional independent vectors.
		Theorem: In a normed linear vector space, any finite-dimensional subspace is complete and hence forms a Banach space.
	web.ics.purdue.edu /~nowack/geos657/lecture3-dir/lecture3.htm (881 words)

	PlanetMath: vector subspace
		Every vector space is a vector subspace of itself.
		Cross-references: finite dimensional, orthogonal complement, inner product space, linear mapping, intersection, sum, vector, necessary and sufficient, subset, field, vector space
		This is version 15 of vector subspace, born on 2001-10-29, modified 2007-02-28.
	planetmath.org /encyclopedia/ProperVectorSubspace.html (96 words)

	Linear Algebra Done Right Preface to Instructor
		To prove the theorem about existence of eigenvalues on complex vector spaces, most books must define determinants, prove that a linear map is not invertible if and only if its determinant equals 0, and then define the characteristic polynomial.
		Inner-product spaces are defined in Chapter 6, and their basic properties are developed along with standard tools such as orthonormal bases, the Gram-Schmidt procedure, and adjoints.
		The spectral theorem, which characterizes the linear operators for which there exists an orthonormal basis consisting of eigenvectors, is the highlight of Chapter 7.
	www.axler.net /LADRPrefaceInstructor.html (1158 words)

	Linear Algebra
		Mathematicians have generalized the definition of a vector space: a general vector space has the properties we’ve listed above for three-dimensional real vectors, but the operations of addition and multiplication by a number are generalized to more abstract operations between more general entities.
		To go from the familiar three-dimensional vector space to the vector spaces relevant to quantum mechanics, first the real numbers (components of the vector and possible multiplying factors) are to be generalized to complex numbers, and second the three-component vector goes an
		This notion naturally extends to vectors and numbers: the adjoint of a ket is the corresponding bra, the adjoint of a number is its complex conjugate.
	galileo.phys.virginia.edu /classes/751.mf1i.fall02/751LinearAlgebra.htm (2711 words)

	Science Fair Projects - List of mathematical theorems
		In some fields, theorem can be considered as a courtesy title, given to major results, although with a content that would not satisfy a mathematician.
		Most of the results do come from mathematics, but there are others from theoretical physics, economics and so on.
		Dimension theorem for vector spaces (vector spaces, linear algebra)
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/List_of_mathematical_theorems (337 words)

	APPENDIX B
		The Lie algebra gl(L), of GL(L) is then the vector space of all endomorphisms of L equipped with a bracket operation defined as commutator of the endomorphisms.
		The Cartan metric metric provides a map between L and its dual space L^+, the space of linear functionals acting on L. The adjoint action of the group on L is then mapped by the Cartan metric to an action of the group on L^+, the coadjoint representation.
		The dimensions of the GAMMA_k are 2^(nu) for n = 2(nu) and for n = 2(nu) + 1.
	graham.main.nc.us /~bhammel/FCCR/apdxB.html (7012 words)

	SparkNotes: Introduction to Vectors: Introduction to Vectors
		Technically speaking, a vector is defined as an element of a vector space, but since we will only be dealing with very special types of vector spaces (namely, two- and three-dimensional Euclidean space) we can be more specific.
		The situation for three-dimensional vectors is very much the same, with an ordered triplet (a, b, c) being represented by an arrow from the origin to the corresponding point in three-dimensional space.
		For instance, in quantum mechanics vectors often come in the form of functions (for instance, a particle wave function), and in such a case it doesn't make sense to talk about the "direction" of the vector.
	www.sparknotes.com /physics/vectors/intro/section1.html (404 words)

	Theory Continued page 3
		While the value of a form (when a vector is fed to it) is typically a scalar, a composite object (form or vector or matrix) could also be the value.
		More useful in higher dimensions and lumpy spaces is, if an equal displacement is made along each path (each according to its own parameter), the distance between the path points is stationary in the sense that its second derivative is zero versus the (equal) parameters at x1 and x2.
		The covariant derivative of the vector field measures the vectors' deviation from parallelism, and is zero at x2 when the field is parallel at x2.
	www.superstringtheory.fanspace.com /custom4.html (3489 words)

	Vector Spaces, Bases, and Dimension
		A basis of a vector space V can be thought of as the same thing -- a set of vectors that define co-ordinate axes, such that every vector can be written as a unique combination of the basis vectors.
		is a vector space of dimension n -- we know the standard basis has n elements -- the equivalence of (2),(3),(4) and the equivalence of (5),(6),(7) follow immediately from Corollary 7.17.
		We proved this is impossible in the finite case of Theorem 7.15.
	www.uwm.edu /~adbell/Teaching/631/1999/631notes7L/node1.html (3461 words)

	Rank
		A common theme of matrices has been viewing a matrix as either a collection of row vectors or a collection of column vectors.
		Of particular interest is the dimension of the row space.
		vector in a basis for the row space of
	www.ltcconline.net /greenl/courses/203/Vectors/rank.htm (413 words)

	Linear Algebra (Math 2318) - Vector Spaces - Basis and Dimension (Site not responding. Last check: 2007-10-13)
		In this section we’re going to take a look at an important idea in the study of vector spaces. We will also be drawing heavily on the ideas from the previous two sections and so make sure that you are comfortable with the ideas of span and linear independence.
		If this system has only the trivial solution the vectors will be linearly independent and if it has solutions other than the trivial solution then the vectors will be linearly dependent.
		Likewise, in Example 1(c) from the section on Linear Independence we saw that these vectors are linearly independent.
	tutorial.math.lamar.edu /AllBrowsers/2318/Basis.asp (856 words)

	Bob Gardner's "Vector Space Book" Prospectus
		The proposed book would be a textbook on vector spaces aimed at the sophomore level university student.
		This is the Riesz-Fisher Theorem and will require the reiteration of some calculus concepts (such as limits in a metric space) but the necessary background will be provided and all results on this topic will be given mathematically rigorous proofs.
		A final section will address the question "What is our universe: a vector space or a manifold?" The idea of the global topology of a manifold will be discussed and potential experiments will be mentioned which may determine this property for our universe.
	www.etsu.edu /math/gardner/vs/prospectus.htm (724 words)

	Dimension (Site not responding. Last check: 2007-10-13)
		Well they are all linearly independent and they all span the vector space by definition.
		Theorem 2.14 says that they also have the same number of elements, or the same length.
		This allows us to define dimension as the number of elements in a basis.
	www.rpi.edu /~piperb/linalg/lecture/09-21/node2.html (57 words)

	Real Algebraic and Analytic Geometry - Preprint Server
		Andreas Bernig, Alexander Lytchak: Tangent spaces and Gromov-Hausdorff limits of subanalytic spaces.
		Riccardo Ghiloni: Explicit Equations and Bounds for the Nakai--Nishimura--Dubois--Efroymson Dimension Theorem.
		Aleksandra Nowel, Zbigniew Szafraniec: On trajectories of analytic gradient vector fields on analytic manifolds.
	www.uni-regensburg.de /Fakultaeten/nat_Fak_I/RAAG (2265 words)

	Colin Rourke's WWW Homepage
		We define flows which straighten vector fields and which then allow a given embedding or immersion to be `compressed' to an immersion in a lower dimension.
		This is a rewritten version of the last paper and includes the underlying classification theorem for links in terms of the rack and the canonical class in \pi_2 of the rack space.
		Markov's theorem for 3-manifolds by Sofia Lambropoulou and Colin Rourke
	www.maths.warwick.ac.uk /~cpr (2010 words)

	WWW interactive multipurpose server
		OEF vector space definition, collection of exercices on the definition of vector spaces.
		OEF subspace definition, collection of exercices on the definition of subspace of vector spaces.
		Extend-subspace, extend a vector subspace to a required dimension.
	www.eval-wims.com /wims (2312 words)

	Double duals of finite dimensional vector spaces.
		The statement of the theorem does not involve dimension, except in the innocuous looking qualification that V should be finite-dimensional, so one might think that there was a way of doing things more canonically, and avoiding the unpleasantly arbitrary choice of basis.
		If the vector space V is infinite-dimensional, then this means that V contains a subset B such that every vector v in V is a linear combination of (finitely many) elements of B, and any finite subset of B is linearly independent in the usual sense.
		This shows that the natural embedding is not an isomorphism, and that is enough to indicate that there will not be an easy proof for finite-dimensional vector spaces (since such a proof would use the natural embedding).
	www.dpmms.cam.ac.uk /~wtg10/meta.doubledual.html (1738 words)

	Euler's Formula
		We interpret the subsets of faces of dimension i as a vector space over the two-element field GF(2), with vector addition being performed by symmetric difference of subsets (also known as exclusive or).
		The sets of faces of the polytope can be interpreted as forming bases for these vector spaces.
		If v is a basis vector, corresponding to a single face of the polytope, this is true because any ridge (face two dimensions lower than v) forms the boundary between two facets of v, and is therefore cancelled out in the calculation of b(b(v)).
	www.ics.uci.edu /~eppstein/junkyard/euler/binary.html (642 words)

	GeoSci 236: The Fundamental Theorem of Linear Algebra
		GeoSci 236: The Fundamental Theorem of Linear Algebra
		To enable visualization, both the range and the domain spaces are depicted as three-dimensional.
		You might find it puzzling (or overly restrictive) that the 2 spaces are presented as aligned with the principal coordinates.
	geosci.uchicago.edu /~gidon/geosci236/fundam (618 words)

	2007-2008 Faculty of Arts and Science Calendar
		The general formulation of non-relativistic quantum mechanics based on the theory of linear operators in a Hilbert space, self-adjoint operators, spectral measures and the statistical interpretation of quantum mechanics; functions of compatible observables.
		The basic concepts of calculus: limits and continuity, the mean value and inverse function theorems, the integral, the fundamental theorem, elementary transcendental functions, Taylor’s theorem, sequence and series, uniform convergence and power series.
		Differential and integral calculus of vector valued functions of a vector variable, with emphasis on vectors in two and three dimensional euclidean space.
	www.artsandscience.utoronto.ca /ofr/calendar/crs_mat.htm (2682 words)

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