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Topic: Dimension of a vector space


  
  Vector space - Wikipedia, the free encyclopedia
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra.
Given a vector space V, any nonempty subset W of V which is closed under addition and scalar multiplication is called a subspace of V. It is easy to see that subspaces of V are vector spaces (over the same field) in their own right.
A vector space with a topology compatible with the operations (i.e., such that addition and scalar multiplication are continuous maps) is called a topological vector space.
www.wikipedia.com /wiki/vector+space   (1071 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/Dimension_(vector_space)   (471 words)

  
 Vector space -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-06)
A vector space (or linear space) is the basic object of study in the branch of (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics called (The part of algebra that deals with the theory of linear equations and linear transformation) linear algebra.
A vector space over the field of (A number of the form a+bi where a and b are real numbers and i is the square root of -1) complex numbers C is called a complex vector space.
Given two vector spaces V and W over the same field F, one can define (Click link for more info and facts about linear transformation) 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.absoluteastronomy.com /encyclopedia/v/ve/vector_space.htm   (1260 words)

  
 PlanetMath: dimension (vector space)
Note: in circumstances where the choice of field is ambiguous, the dimension of a vector space depends on the choice of field.
For example, every complex vector space is also a real vector space, and therefore has a real dimension, double its complex dimension.
This is version 5 of dimension (vector space), born on 2002-06-01, modified 2004-03-04.
planetmath.org /encyclopedia/Dimension2.html   (108 words)

  
 Dimension (vector space) -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-06)
It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of (The magnitude of something in a particular direction (especially length or width or height)) dimension.
All bases of a vector space have equal cardinality (see (Click link for more info and facts about dimension theorem for vector spaces) dimension theorem for vector spaces) and so the Hamel dimension of a vector space is uniquely defined.
One can see a vector space as a particular case of a (Click link for more info and facts about pregeometry) pregeometry, and in the latter there is a well defined notion of dimension.
www.absoluteastronomy.com /encyclopedia/d/di/dimension_(vector_space).htm   (685 words)

  
 Vector space   (Site not responding. Last check: 2007-11-06)
Given a vector space V, any nonempty subset W of V which is closed under addition and scalar multiplication is called a subspace of V. It is easy to see that subspaces of Vare vector spaces (over the same field) in their own right.
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 arecompatible with the relevant structure—i.e., they preserve sums and scalar products.
The mostbasic physical vector is the displacement vector from point A topoint B (its direction is from A to B and its length is the distance between A and B).
www.therfcc.org /vector-space-3889.html   (1079 words)

  
 Vector space - InfoSearchPoint.com   (Site not responding. Last check: 2007-11-06)
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.infosearchpoint.com /display/Linear_space   (850 words)

  
 [No title]
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.ercangurvit.com /vectorspaces/vecspaceeng.htm   (4069 words)

  
 Cayley-Dickson construction - Wikipedia, the free encyclopedia
Besides being of higher dimension, the complex numbers can be said to lack one algebraic property of the real numbers: a real number is its own conjugate.
Inasmuch as quaternions consist of two independent complex numbers, they form a 4-dimensional vector space.
Inasmuch as octonions consist of two quaternions, the octonions form an 8-dimensional vector space.
en.wikipedia.org /wiki/Cayley-Dickson_construction   (737 words)

  
 VECTOR SPACE
A subspace of a vector space is a vector space that is inside another vector space.
The dimension of a vector space is simply the number of basis vectors needed to define it.
If the vector x is represented by (1,0,0), the vector y as (0,1,0) the vector z as (0,0,1), and the vector 3x as (3,0,0) the resulting matrix would have the RANK of 3 because the 4 vectors only span a 3 dimensional space.
www.qsl.net /k9so/VECTORSPACE.htm   (1529 words)

  
 GAP Manual: 32. Vectors   (Site not responding. Last check: 2007-11-06)
A vector space V is a set of vectors, for which an addition u + v and a multiplication by scalars, i.e., elements from F, s v must be defined.
This is a little bit lax, because the dimension is a property of the vector space, not of the vector, but should seldom cause confusion.
Vectors play an important role for matrices (see chapter Matrices), which are implemented as lists of vectors.
www.math.uiuc.edu /Software/GAP-Manual/Vectors.html   (482 words)

  
 GAP Manual: 9 Vector Spaces   (Site not responding. Last check: 2007-11-06)
A special type of vector space, that is implemented in the GAP library, handles the case where the elements Row Spaces for details).
The components listed in Vector Space Records are common for all vector spaces, but special types of vector spaces, such as the row spaces, may use additional entries to store specific data.
The intersection of vector spaces is computed by finding a base for the intersection of the sets of their elements.
www.mcs.kent.edu /system/documentation/gap/CHAP009.htm   (1060 words)

  
 Rank (linear algebra) - Wikipedia, the free encyclopedia
The maximal number of linearly independent columns of the m-by-n matrix A with entries in the field F is equal to the dimension of the column space of A (the column space being the subspace of F
then the rank of A can also be defined as the dimension of the image of f, or as n minus the dimension of the kernel of f (see linear map for a discussion of image and kernel).
In those generalisations, column rank, row rank, dimension of column space and dimension of row space of a matrix may be different from the others or may not exist.
www.wikipedia.org /wiki/Rank_of_a_matrix   (730 words)

  
 Hamel dimension : Dimension of a vector space   (Site not responding. Last check: 2007-11-06)
The dimension of a vector space V is the cardinality (i.e.
It is sometimes called Hamel dimension when it is necessary to distinguish it from other types of dimension.
However, the Hamel dimension depends on the base field, so while R has dimension 1 when considered as a vector space over itself, it has dimension c (the cardinality of the continuum) when considered as a vector space over Q (the rationals).
www.city-search.org /di/dimension-of-a-vector-space.html   (277 words)

  
 The SUIFMATH Library - Vector Space Utilities   (Site not responding. Last check: 2007-11-06)
method increases the space dimension of the vector space to the value given as its argument.
The argument is factored with the LU decomposition of the vector space's matrix representation.
If the resulting factored vector is dependent, then it is already in the vector space.
suif.stanford.edu /suif/suif1/docs/suifmath_30.html   (159 words)

  
 Basis 4
The dimension of a vector space is the number of vectors in a basis.
From this exercise, we see that a vector space is finite dimensional if and only if it can be spanned by finitely many vectors.
Thus the basis is not unique, and the definition of dimension must be justified by showing that different choices of bases will always have the same number of vectors.
algebra.math.ust.hk /vector_space/08_basis/lecture4.shtml   (220 words)

  
 Clearing up the market cycle... best Quotient Vector Space   (Site not responding. Last check: 2007-11-06)
The dimension of the quotient vector space is called the codimension of relative to.
dimension (vector space) dimension (vector space) Let V be a vector space over a field K. We say that V is finite-dimensional if there exists a finite basis of V. Otherwise we call V infinite-dimensional.
In functional analysis, a seminorm is a function on a vector space with certain properties characteristic of a measure of "length".
ascot.pl /th/Fourier5/Quotient-Vector-Space.htm   (479 words)

  
 is there a proof for "{} is an orthonormal basis for any zero-dimensional vector space"?   (Site not responding. Last check: 2007-11-06)
The span of the empty set is the collection of all finite sums of scalar multiples of vectors in the empty set; that is, the span of the empty set consists exactly of the empty sum.
By definition, the span of the empty set is {0}; by vacuity, the empty set is linearly independent (if it were not independent, there would be a vector in the empty set which is a linear combination of the previous vectors, which is certainly not the case).
Any linearly independent set of vectors is a basis for its span, so the empty set is a basis for the zero-dimensional vector space {0}.
www.thehelparchive.com /new-2374331-279.html   (2731 words)

  
 Basis and Dimension
The row space of an mxn matrix A is the subspace of R
The column space of A is the subspace of R
spanned by the column vectors of A. The dimension of the row space and the dimension of the column space are equal (Theorem 5.12) and are defined to be the rank of the matrix A, which will always be the number of nonzero rows in the row-reduced echelon form of A (Theorem 5.13).
www.saintjoe.edu /~karend/m244/m244-Basis.html   (414 words)

  
 Hamel dimension   (Site not responding. Last check: 2007-11-06)
It is sometimes called Hamel dimension to distinguish it from other types of dimension.
All bases of a vector space have equal cardinality (see dimension theorem for vector spaces) and so the Hamel dimension of a vector space is uniquely defined.
The length of a module and the rank of an abelian group both have several properties similar to the Hamel dimension of vector spaces.
www.sciencedaily.com /encyclopedia/hamel_dimension   (472 words)

  
 [No title]   (Site not responding. Last check: 2007-11-06)
We say V is finite-dimensional if it is spanned by a finite set of vectors, and in this case, the number of elements in any basis is the dimension of V. If V is not spanned by a finite set of vectors, then V is infinite-dimensional.
Finally, we discussed the Basis Theorem, which says that if a vector space has dimension n, then a set of n vectors which is linearly independent must be a basis, and a set of n vectors which spans must be a basis.
This means that if you know the dimension of a vector space and you want to check if a set of vectors is a basis, then you only have to check one of the two properties for a basis (linear independence OR spanning), not both.
www.mathcs.emory.edu /~vicki/221/diary/nov28.html   (165 words)

  
 The Steinitz Theorem and the Dimension of a Vector Space
The Steinitz Theorem and the Dimension of a Vector Space
The main purpose of the paper is to define the dimension of an abstract vector space.
The dimension of a finite-dimensional vector space is, by the most common definition, the number of vectors in a basis.
www.mizar.org /JFM/Vol7/vectsp_9.html   (219 words)

  
 Vector Spaces, Bases, and Dimension
When R is a field, we use the name vector space  instead of module.
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.
www.uwm.edu /~adbell/Teaching/631/1999/631notes7L/node1.html   (3461 words)

  
 Emergence-Based Approach to Categorization and Similarity
As yet the vector space approach is the only example we have of a broadly defined, precisely conceived approach to similarity and categorization.
A metric on multidimensional space is then used to determine the distance between texts; this is also simple enough, though there is a serious and largely arbitrary decision to be made as regards the weighting of differences between words versus differences between phrases of different lengths.
The distance between two vectors, according to standard vector space metrics, is not the same as the extent to which two texts enhance one another's compressibility.
www.goertzel.org /papers/catpap.html   (5758 words)

  
 Extract 8
Context: The Extract presented here follows Extract 8.4 where the notions of a basis and the dimension of a vector space were exemplified in finding the dimension of the vector spaces given in LA6.26.
Y),where X and Y are subspaces of a vector space V, which he remembers, he says, from his reading, not the lectures.
The tutor asks them to consider the information they have: X and Y are of dimension n-1 in a space of dimension n.
www.uea.ac.uk /~m011/thesis/appendices/appendices8/8v.htm   (476 words)

  
 Hamel dimension : Dimension (linear algebra)   (Site not responding. Last check: 2007-11-06)
The dimension of a vector space V is the cardinality (i.
Every basis of a vector space has equal cardinality and so the Hamel dimension of a vector space is uniquely defined.
is a vector space of dimension n over R (the reals).
www.termsdefined.net /di/dimension-(linear-algebra).html   (372 words)

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