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Topic: Vector subspace

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	PlanetMath: closure of a vector subspace is a vector subspace
		PlanetMath: closure of a vector subspace is a vector subspace
		closure of a vector subspace is a vector subspace
		This is version 5 of closure of a vector subspace is a vector subspace, born on 2005-02-04, modified 2005-03-01.
	planetmath.org /encyclopedia/ClosureOfAVectorSubspaceIsAVectorSubspace2.html (163 words)

	PlanetMath: vector subspace
		Every vector space contain two trivial vector subspaces: the entire vector space, and the zero vector space.
		If S and T are vector subspaces of a vector space V, then the vector sum
		This is version 10 of vector subspace, born on 2001-10-29, modified 2005-07-20.
	planetmath.org /encyclopedia/Subspace.html (117 words)

	Dictionary of Meaning www.mauspfeil.net
		Vector addition is just matrix addition and scalar multiplication is defined in the obvious way (by multiplying each entry by the same scalar).
		An important example arising in the context of linear algebra itself is the vector space of linear transformations.
		This is a vector subspace of all possible real-valued functions on ''X'' since the sum of any two continuous functions is continuous and scalar multiplication is continuous.
	www.mauspfeil.net /Examples_of_vector_spaces.html (1402 words)

	Subspace_Basis.htm (Site not responding. Last check: 2007-09-11)
		A subset of vectors is said to be "linearly independent" if and only if the only linear combination of those vectors that produces the zero vector is the one where all of the c's are zeros.
		A subset of vectors from a vector space is said to be a "basis" for that vector space if and only if the set of these vectors both span the space and is linearly independent.
		Writing a vector as a linear combination of a set of vectors turns out to be equivalent to a matrix algebra problem and/or a system of equations problem.
	www.hvcc.edu /~murrajoh/caliii/linal/subspacebasis/subspacebasis.htm (1841 words)

	Generating a vector space
		Given a vector space V over a field K, a subset W of V is said to generate (or span) V if every element of V can be expressed as a linear combination of elements of W.
		Let U and W be subspaces of the vector space V over K.
		Then U+W, called the sum of U and W, is a vector subspace of V and is spanned by the union of U and W.
	www.ebroadcast.com.au /lookup/encyclopedia/ge/Generating_a_vector_space.html (248 words)

	Dictionary of Meaning www.mauspfeil.net
		The concept of a '''linear subspace''' (or '''vector subspace''') is important in linear algebra and related fields of mathematics.
		A linear subspace is usually called simply a ''subspace'' when the context serves to distinguish it from other kinds of subspaces.
		Conditions 1, 2, and 3 for a subspace are simply the most basic kinds of linear combinations (where for condition 3, we remember that a linear combination of no vectors at all yields the zero vector).
	www.mauspfeil.net /Linear_subspace.html (1039 words)

	Patent 5634199: Method of subspace beamforming using adaptive transmitting antennas with feedback
		The method of claim 5, wherein each diversity vector is a dominant generalized eigenvector of a pair of matrices, where each matrix of the pair is formed from a linear combination of the autocorrelation matrices.
		Once the diversity vectors are calculated, probing and feedback are used by the transmitter to measure a property of the instantaneous channel vector.
		The subspace is the set of directions in which the channel vector tends to point.
	www.freepatentsonline.com /5634199.html (6015 words)

	- US Patent (Site not responding. Last check: 2007-09-11)
		Procedures according to claim 1, characterized in that after successful detection, the detected symbol vector is subtracted from the received signal matrix X, Y or A in the order of decreasing signal power, estimated using signal matching filtering (correlation of the known training sequences with the received data), before the next symbol vector is detected.
		Procedures according to claim 6, characterized in that after successful detection, the detected symbol vector is subtracted from the received signal matrix X, Y or A in the order of decreasing signal power, estimated using signal matching filtering (correlation of the known training sequences with the received data), before the next symbol vector is detected.
		Additionally simultaneous adaptation of the symbol vector to the vector subspace and to the symbol quantity of finite size takes place by means of the least squares error algorithm.
	www.patentstorm.us /patents/6829312.html (3652 words)

	QUANTUM UNIVERSE AS ALGEBRA
		The multiplicity of sharing of subspaces, which is the uniform weight of any vertex of the geometric simplex as graph (a complete graph) is a dimension of a simplex.
		The dimension of the algebra is the sum of the dimensions of these subspaces, which sums exactly the same as for the power set S, and the quantization of the family of subsets of cardinality m is the algebra subspace spanned by m-fold products of the generators.
		The sharing of subspaces is a sharing of subspaces of both carrier spaces and of the algebras.
	graham.main.nc.us /~bhammel/PHYS/algu.html (11111 words)

	PlanetMath: vector space
		is the zero vector in a vector space
		This is version 11 of vector space, born on 2001-10-19, modified 2005-02-23.
		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/LinearSpace.html (263 words)

	Patent 6006110: Wireless communication network using time-varying vector channel equalization for adaptive spatial
		As is described herein, statistical characterization of the receive channel vector provides information which may be used by the base station to determine an "optimal" transmit beam pattern, i.e., a transmit beam pattern which maximizes the average signal-to-interference power delivered to a given mobile user.
		3, a vector channel covariance estimator 140 within the transmit beamformer 64 operates to produce a statistical characterization of a receive channel vector using: (i) the outputs x.sub.R,m (k), m=1 to M, of the complex frequency downconverter 112, and (ii) an estimate of the desired signal S.sub.r (k) generated in the receive channel beamformer 60.
		The fundamental contributions are the blind SINR adaption criteria, the subspace decomposition and the use of correlated desired channel behavior to improve estimation accuracy for the desired channel and the interference subspace.
	www.freepatentsonline.com /6006110.html (7685 words)

	Subspace 1
		= {polynomials of degree ≥ 3 and ≤ 5} is not a subspace of
		The following shows that a (infinite) straight line passing through the origin is a subspace, while the one not passing through the origin is not a subspace.
		A subspace, with the inherited operations, is a vector space.
	algebra.math.ust.hk /vector_space/02_subspace/lecture1.shtml (421 words)

	PlanetMath: annihilator of vector subspace (Site not responding. Last check: 2007-09-11)
		"annihilator of vector subspace" is owned by stevecheng.
		Cross-references: pullback, image, linear operator, sum, theorem, dimension, natural isomorphism, finite-dimensional, vector, dual space, subspace, subset, vector space
		This is version 2 of annihilator of vector subspace, born on 2005-07-30, modified 2005-07-30.
	planetmath.org /encyclopedia/Annihilator4.html (68 words)

	Ex Astris Scientia - Treknology Encyclopedia A-K
		The unit cochrane is employed to measure the power of a warp drive as well as of other devices which create a subspace field, such as the impulse drive and the FTL computer core.
		Personnel subspace communication device, originally hand-held, later in the 24th century integrated in the Starfleet badge; the latter is also referred to as comm badge.
		The STTNG Technical Manual, however, states that the subspace driver coil was introduced as late as in the Ambassador class, although it must have existed earlier according to Krauss's calculations.
	www.ex-astris-scientia.org /treknology1.htm (7234 words)

	Vector Subspace test on polynomials (Site not responding. Last check: 2007-09-11)
		Yet if the answer sheet is saying that p3(x) IS in the subspace (where the sum of the coefficiants of p3(x) = 0), then I apparently need a crash course in polynomial addition.
		Why the answer sheet is treating this as a function space rather than a polynomial space is beyond my comprehension.
		Determine if the following is a subspace of P3 (polynomials of degree 3 or less): Set of polynomials a0 + a1x + a2x^2 + a3x^3 for which a0 + a1 + a2 + a3 = 0 I got this problem wrong.
	www.thehelparchive.com /new-2369547-279.html (417 words)

	Reducing Vectors Relative to a Subspace (Site not responding. Last check: 2007-09-11)
		(Function.) Given a vector v from a tuple module V and a submodule W of V, return the reduction of v with respect to W (that is, the canonical representative of the coset v + W).
		(Procedure.) Given a vector v from a tuple module V and a submodule W of V, replace v with its reduction of with respect to W (that is, the canonical representative of the coset v + W).
		Given a vector v from a tuple module V and a submodule U of V, return the unique u in U and w in the complement to U in U + such that v = u + w.
	www.dtr.isy.liu.se /Magma/text597.html (120 words)

	Genome Biology \| Full text \| Vector algebra in the analysis of genome-wide expression data
		It can easily be seen that with respect to the data matrix, a gene vector consists of the elements of a row and an experiment vector consists of the elements of a column (Figure 1a).
		Vector angle is the counterpart to the Pearson correlation coefficient
		vectors from SVD (which are linear combinations of the experiment vectors) it would be useful to identify which actual experiment vectors are most useful to form a basis for the matrix.
	genomebiology.com /2002/3/3/research/0011 (6658 words)

	Subspaces of 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
		of the subspaces is a subspace of the vector space
	www.cs.ut.ee /~toomas_l/linalg/lin1/node6.html (323 words)

	Subspace 2
		The sum and intersection of subspaces are still subspaces.
		We note that the sum and the intersection are the adaptations of the union and the intersection of sets to vector spaces.
		While the intersection remains the same, the sum is different form the union, because the union of two subspaces is not a subspace.
	algebra.math.ust.hk /vector_space/02_subspace/lecture2.shtml (149 words)

	Linear Algebra Vector Space/Subspace Span debacle (Site not responding. Last check: 2007-09-11)
		All bases consist of two vectors - that is the _definition_ of dim = 2.
		This = vector space/subspace/span/basis seems to be so very much abstract that = reading these theorems in the books (without decent examples) leaves me = more confused and bewildered after reading it.
		This vector space/subspace/span/basis seems to be so very much abstract that reading these theorems in the books (without decent examples) leaves me more confused and bewildered after reading it.
	www.thehelparchive.com /new-2366726-279.html (3414 words)

	Vector spaces and vector subspaces.
		This subset is called a vector subspace of V if it is a vector space, when the addition and multiplication by scalars are the restriction to W of those of V.
		The other properties are trivially fulfilled, excepted the belonging to W of the zero vector, which is a consequence of the second requirement.
		Proposition 5.1.5 The subset W of V is a (vector) subspace of V if, and only if, it is closed under linear combinations.
	ndp.jct.ac.il /tutorials/alg-tut/node16.html (273 words)

	[No title] (Site not responding. Last check: 2007-09-11)
		Sometimes, it's more convenient to have a list of basis vectors, sometimes an LU decomposition of a matrix whose kernel is the vector space of interest.
		Typically, we use this to represent a "vector space" over integers, but in by the definition of vector space there is no such thing, and the mathematics breaks down for integers.
		So the basis vector matrix, in which each row of the array is a basis vector, is D() x N().
	www-rocq.inria.fr /~pop/doc/lno/vs.txt (361 words)

	Application of Vector-Valued Rational Approximations to the Matrix Eigenvalue Problemand Connections with Krylov ...
		These generalizations can be used to obtain simultaneously several of the largest distinct eigenvalues and corresponding eigenvectors and other vectors in the invariant subspaces.
		In addition, it is shown that the generalized power methods of this work are equivalent to some Krylov subspace methods, among them the methods of Arnoldi and Lanczos.
		At the same time this theory suggests a new mode of usage for these Krylov subspace methods that has been observed to possess computational advantages over their common mode of usage in some cases.
	epubs.siam.org /sam-bin/dbq/article/24316 (394 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.
		In terms of our vector analogy, the subspace spanned by a set of vectors is the smallest linear space through the origin containing all the vectors.
		One way to avoid this problem is to prove all results only for finitely spanned vector spaces, that is, vector spaces which have a finite spanning set.
	www.uwm.edu /~adbell/Teaching/631/1999/631notes7L/node1.html (3461 words)

	Data assimilation concepts and methods
		How the vector components relate to the real state depend on the choice of discretization, which is mathematically equivalent to a choice of basis.
		The key to data analysis is the use of the discrepancies between observations and state vector.
		According to the previous paragraph, this is given by the vector of departures at the observation points:
	www.ecmwf.int /newsevents/training/rcourse_notes/DATA_ASSIMILATION/ASSIM_CONCEPTS/Assim_concepts3.html (614 words)

	Linear Systems
		There are comparison maps: one to produce the vector of coefficients of polynomials with respect to the given basis and one to create a polynomial from a vector of coefficients.
		The correspondence between vectors of coefficients and polynomials are computed at the level of the complete systems so that any two subsystems interpret coefficient vectors with respect to the same basis of polynomials.
		The subsystem of the linear system L determined by the subspace V of the complete coefficient space of L. It is an error to call this if V is not a subspace of the coefficient space of L. Basic Algebra of Linear Systems
	www.math.lsu.edu /magma/text1143.htm (3365 words)

	Linear Dependence of Vectors. Basis of the Vector Space.
		Prove that each set of vectors that contains the null vector is linearly dependent.
		is not expressable as a linear combination of vectors of V.
		A subset B of the vectors of the vector space
	www.cs.ut.ee /~toomas_l/linalg/lin1/node7.html (377 words)

	NSDL Metadata Record -- zero vector space
		Definition A zero vector space is a vector space that contains only one element, a zero vector.
		Every vector space has a zero vector space as a vector subspace.
		A vector space X is a zero vector space if and only if the dimension of X is zero.
	nsdl.org /mr/1034916 (69 words)

	icpr,17th International Conference on Pattern Recognition (ICPR'04) - Volume 2 (Site not responding. Last check: 2007-09-11)
		In this paper, we present a novel maximum correlation sample subspace method and apply it to human face detection [Detecting Face in Images: A Survey] in still images.
		After a base vector is selected, all other samples are made orthogonal to the current base vector and which is in turn used to form the training samples for learning the next base vector.
		Each subspace base is created by a one-pass process and therefore the method is computationally very efficient.
	csdl.computer.org /comp/proceedings/icpr/2004/2128/02/212820423abs.htm (250 words)

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