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Topic: Vector space

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  Vector space - Wikipedia, the free encyclopedia
In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors.
Vector spaces are the basic objects of study in linear algebra, and are used throughout mathematics, the sciences, and engineering.
Vectors in these spaces are ordered pairs or triples of real numbers, and are often represented as geometric vectors (quantities with a magnitude and a direction, usually depicted as arrows).
en.wikipedia.org /wiki/Vector_space   (1420 words)

 Topological vector space - Wikipedia, the free encyclopedia
As the name suggests the space blends a topological structure (a uniform structure to be precise) with the algebraic concept of a vector space.
A vector space is an abelian group with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as multiplication by −1).
Vector addition and scalar multiplication are not only continuous but even homeomorphic which means we can construct a base for the topology and thus reconstruct the whole topology of the space from any local base around the origin.
en.wikipedia.org /wiki/Topological_vector_space   (1065 words)

 Online Encyclopedia and Dictionary - Vector space
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra.
The concept of a vector space is entirely abstract, like the concepts of a group, ring, and field.
Given two vector spaces V and W over the same field F, one can define linear transformations or “linear maps” from V to W. These are maps from V to W which are compatible with the relevant structure—i.e., they preserve sums and scalar products.
www.fact-archive.com /encyclopedia/Vector_space   (1018 words)

 [No title]   (Site not responding. Last check: 2007-10-21)
The fundamental concept in linear algebra is that of a vector space or linear space.
It is a generalization of the set of all geometrical vectors and is used throughout modern mathematics.
Given two vector spaces V and W over the same field, one can define linear transformations or "linear maps" from V to W. These are maps from V to W which are compatible with the relevant structure, i.e.
www.wikiwhat.com /encyclopedia/v/ve/vector_space_1.html   (813 words)

 Vector Space -- Recommendations and Resources   (Site not responding. Last check: 2007-10-21)
In linear algebra and related areas of mathematics, the null vector or zero vector in a vector space is the uniquely-determined vector, usually written 0, that is the identity element for vector addition.
The * was used in the vector space axioms both as a map * : F x F -> F and as a map * : F x V -> V. I changed a*b to ab.
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e.
www.becomingapediatrician.com /health/156/vector-space.html   (1126 words)

 Linear Vector Spaces   (Site not responding. Last check: 2007-10-21)
The space of ordinary vectors in three-dimensional space is 3-dimensional.
is a subspace of the space of ordinary vectors in 3 dimensions.
Vector addition is different from ordinary addition, but it obeys the rules for the addition operation of a vector space.
electron6.phys.utk.edu /qm1/modules/m3/Vector_space.htm   (1211 words)

 Finite Vector Space Representations   (Site not responding. Last check: 2007-10-21)
The vector representation of fields, and the matrix representation of linear operators is derived, together with how vectors and matrices are transformed to alternative representations, incuding their eigenrepresentation.
The linear vector space representation of fields and operators is a powerful component in the toolbox of the mathematician, physicist, engineer and statistician.
As in the case of the velocity or force, the vector will have a direction and length, but not in two- or three- dimensional space, but in an abstract -dimensional space, where is the number of grid points in the representation.
www.met.rdg.ac.uk /~ross/DARC/LinearVectorSpaces.html   (843 words)

 PlanetMath: vector space
This is version 11 of vector space, born on 2001-10-19, modified 2005-02-23.
vector space is over division ring by Mathprof on 2006-06-14 15:18:20
Hungerford's _Algebra_ defines vector spaces as unitary modules over division rings, which means funky, noncommutative things can happen (you can have a "left" vector space which isn't a "right" vector space, for example).
planetmath.org /encyclopedia/VectorSpace.html   (274 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)

 Associated Vector Space   (Site not responding. Last check: 2007-10-21)
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 vector space V underlying the space of modular symbols M, the map V -> M, and the map M -> V. DualVectorSpace(M) : ModSym -> ModTupFld
The lattice generated by the integral modular symbols in the vector space representation of the space of modular symbols M. This is the lattice generated by all modular symbols X^iY^(k - 2 - i){a, b}.
magma.maths.usyd.edu.au /magma/htmlhelp/text1339.htm   (161 words)

 Vector Space Tutorial
This vector is a pointer to a place in an N dimensional space where N = 11.
Each term must have its own axis which means that the majority of every vector is going to be empty because most terms are not going to be found in every document.
As an example, in the vector above the term identified by the integer 27 has a word count of 9 in whatever document this vector belongs to.
www.thebananatree.org /vector_space/vector_space.html   (1577 words)

 Topological Vector Space
A topological vector space is a vector space with a topology, such that addition and scaling are continuous.
A normed vector space is a topological vector space, deriving its topology from the metric.
In a metric space, the translate of an open ball is an open ball, since the distance between two points does not change; but in a topological group, we have to use the properties of continuity to prove translation preserves open sets.
www.mathreference.com /top-ban,tvs.html   (1410 words)

 infinite dim vector space
A vector space of dimension n is spanned by a basis of n vectors, just as in your example of R^3.
This is because a basis needs to span the vector space (which means you need *at least* n vectors) and has to be linearly independant (which means you can only have *at most* n vectors) which makes the number of vectors in the basis exactly n.
which is the vector space of all polynomials in x over R. This is trivially an infinite dimensional vector space since a finite number of vectors in a basis contains a vector with a maximum degree r, meaning that x^(r+1) and higher cannot be formed.
www.physicsforums.com /showthread.php?threadid=104094   (460 words)

 Introduction: Vector Space Model   (Site not responding. Last check: 2007-10-21)
The term frequency is somewhat content descriptive for the documents and is generally used as the basis of a weighted document vector [17].
It is also possible to use binary document vector, but the results have not been as good compared to term frequency when using the vector space model [17].
The similarity in vector space models is determined by using associative coefficients based on the inner product of the document vector and query vector, where word overlap indicates similarity.
isp.imm.dtu.dk /thor/projects/multimedia/textmining/node5.html   (667 words)

 Normed Vector Space
A normed vector space, also called a normed linear space, is a real vector space s with a norm function denoted x.
Thus d becomes a distance metric, and s is a metric space, with the open ball topology.
A banach space is a normed vector space that forms a complete metric space.
www.mathreference.com /top-ban,nvs.html   (747 words)

 vector space... help!!!!
Since the definition of vector space is, as you said, a set of vectors together with two operations, this has to be assuming the "standard" operations on R^3: "coordinatewise" addition and scalar multiplication.
In order that a subset of a vector space be a sub-space (a vector space using the same operations) whenever u and v are in the set, u+v must "also" (as well as u and v) be in that same set.
S is not a vector space unless we have the "closure properties" for the operations.
www.physicsforums.com /showthread.php?p=230012   (1518 words)

 PlanetMath: normed vector space
It follows that any normed space is a locally convex topological vector space, in the topology induced by the metric defined above.
norm, metric induced by a norm, metric induced by the norm, normed space
This is version 8 of normed vector space, born on 2002-01-24, modified 2006-06-04.
planetmath.org /encyclopedia/NormedVectorSpace.html   (257 words)

 Vector Spaces
One of the fundamental concepts of linear algebra is the concept of vector space.
In analysis the notion ``linear space'' is used unstead of the notion ``vector space''.
Instead of the notion ``vector space'' we shall use the abbreviative ``space''.
www.cs.ut.ee /~toomas_l/linalg/lin1/node5.html   (382 words)

 Earliest Known Uses of Some of the Words of Mathematics (V)
This paper adopts the convention of denoting a vector by a single (Greek) letter, and concludes with a discussion of formulae for applying rotations to vectors by conjugating with unit quaternions.
Vector and scalar also appear in 1846 in a paper "On Symbolical Geometry," in the The Cambridge and Dublin Mathematical Journal vol.
In "The Theory of Linear Dependence" Bôcher wrote that "the simplest geometrical interpretation for a complex quantity with n principal units is as a vector in space of n dimensions." (Annals of Mathematics, 2, 1900/1, p.
members.aol.com /jeff570/v.html   (3542 words)

 Abstract linear spaces
The parallel development in analysis was to move from spaces of concrete objects such as sequence spaces towards abstract linear spaces.
Bellavitis then defines the 'equipollent sum of line segments' and obtains an 'equipollent calculus' which is essentially a vector space.
Hamilton represented the complex numbers as a two dimensional vector space over the reals although of course he did not use these general abstract terms.
www-groups.dcs.st-and.ac.uk /~history/HistTopics/Abstract_linear_spaces.html   (1861 words)

 [ref] 58 Vector Spaces
The characteristic (see Characteristic) of a vector space is equal to the characteristic of its left acting domain.
Vector space homomorphisms (or linear mappings) are defined in Section Linear Mappings.
Examples of such vector spaces are vector spaces of field elements (but not the fields themselves) and non-Gaussian row and matrix spaces (see IsGaussianSpace).
www.math.temple.edu /computing/gap/ref/CHAP058.htm   (3619 words)

 Linear Vector Spaces, and Subspaces
Basis is a linearly independent set of vectors which span the vector space.
N is the dimension of the vector space, we can always construct a basis by adding additional independent vectors.
A vector space can be decomposed into independent subspaces instead of vectors.
web.ics.purdue.edu /~nowack/geos657/lecture3-dir/lecture3.htm   (881 words)

 Subspaces of the Vector Space
) that is a vector space will respect to vector addition and multiplication by a number defined in the vector space
To prove sufficiency, we have to show that in our case conditions 1-8 of the vector spaces are satisfied.
Prove that the set of all symmetric matrices form a subspace in the vector space of all square matrices
www.cs.ut.ee /~toomas_l/linalg/lin1/node6.html   (323 words)

 The Classic Vector Space Model
Now we treat weights as coordinates in the vector space, effectively representing documents and the query as vectors.
To find out which document vector is closer to the query vector, we resource to the similarity analysis introduced in Part 2.
A main disadvantage of this and all term vector models is that terms are assumed to be independent (i.e.
www.miislita.com /term-vector/term-vector-3.html   (1153 words)

 vector: Vector Analysis and Vector Space
…, called vectors, for which the operations of addition of vectors and multiplication of a vector by a scalar are defined and which satisfies ten axioms relating to such properties as closure under both operations, associativity, commutativity, and existence of a zero vector, an additive inverse (negative of a vector), and a unit scalar.
A vector space model for variance reduction in single machine scheduling.
Multicategory support vector machines: theory and application to the classification of microarray data and satellite radiance data....
www.infoplease.com /ce6/sci/A0861763.html   (250 words)

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