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Topic: Linear subspace

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	Kids.Net.Au - Encyclopedia > Linear subspace (Site not responding. Last check: 2007-10-10)
		A linear subspace is an important concept 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 subspace.
		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.kids.net.au /encyclopedia-wiki/li/Linear_subspace (614 words)

	NationMaster - Encyclopedia: Quotient space (linear algebra)
		In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero.
		Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations.
		The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics.
	www.nationmaster.com /encyclopedia/Quotient-space-%28linear-algebra%29 (1396 words)

	Affine transformation
		A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations.
		A linear subspace of a vector space is a subset that is closed under linear combinations; an affine subspace is one that is closed under affine combinations.
		The set of linear combinations of a set of vectors is their "linear span" and is always a linear subspace; the set of all affine combinations is their "affine span" and is always an affine subspace.
	www.ebroadcast.com.au /lookup/encyclopedia/af/Affine_map.html (264 words)

	Science Fair Projects - Linear subspace
		A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces.
		That is, W is a subspace iff every linear combination of (finitely many) elements of W also belongs to W.
		Given subspaces U and W of a vector space V, then their intersection U ∩ W := {v ∈ V : v is an element of both U and W} is also a subspace of V.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Linear_subspace (1193 words)

	PlanetMath: vector subspace
		is said to be a vector subspace of
		Every vector space is a vector subspace of itself.
		This is version 15 of vector subspace, born on 2001-10-29, modified 2007-02-28.
	planetmath.org /encyclopedia/Subspace.html (96 words)

	Martina Finzel-Hoffmanns Habilitationsschrift
		The paper investigates classical problems in approximation, the metric projection P of the normed R^n onto a subset G of R^n, corresponding selections, and their construction when G is a linear subspace or a polyhedral subset of R^n.
		Elementary vectors of a subspace are vectors therein which have minimal support.
		If G is a linear subspace the tuples of the classification are exactly the supports of the elementary vectors of the orthogonal complement of G.
	www.uni-giessen.de /www-Numerische-Mathematik/at-net/THESES/finzel.html (621 words)

	Linear subspace - Wikipedia, the free encyclopedia
		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.
		Given subspaces U and W of a vector space V, then their intersection U ∩ W := {v ∈ V : v is an element of both U and W} is also a subspace of V.
	en.wikipedia.org /wiki/Linear_subspace (862 words)

	Prerequisites
		A subspace of V is a nonempty subset W of V that is closed under all operations (so rv is in it if r in F and v in W).
		The linear combinations of a set of vectors X in V form a linear subspace, called the linear subspace of V spanned by X.
		The kernel is a subspace of V, the image is a linear subspace of W.
	www.win.tue.nl /~amc/ow/lba/voorkennisvs.html (595 words)

	MAT 200 Lecture Notes -- Subspaces, Bases, and Dimensions
		A subspace is a subset S of the set of row or column vectors with a given number of entries with the following two properties.
		A basis for a subspace S of row vectors is a set v(1), v(2),...v(m) such that every vector w in S can be written uniquely as a linear combination of v(1), v(2),..., v(m).
		The dimension of a subspace is the number of vectors in a basis for that subspace.
	www.math.princeton.edu /~stalker/200f99/notes_5.html (3537 words)

	Sharipov's linear algebra textbook...
		given a linear map f from R^n to R^m, arrange the image vectors f(e1),...,f(en) as columns in a matrix M. then the columns of M have length m and there are n of them, i.e.
		If f is any linear map from V to W, then by choosing bases for V and W we obtain isomorphisms between them and some R^n and R^m, hence we obtain a resulting map from R^n to R^m which has a matrix.
		But th tangent space at x is the subspace of vectors pwerpendicular to x, and the gradient of f at x is the vector 2Ax.
	www.physicsforums.com /showthread.php?t=58313&page=1&pp=15 (4031 words)

	Research Statement
		Instead we proposed using a universal subspace for overcoming generalization/over-fitting problem in applications such as face recognition [4], Combining subspace and discriminant analysis, we proposed a general framework to solve practical classification problems.
		For such case, multiple subspaces or parametric subspace can be constructed from the original subspace to accommodate the inputs distorted by scaling, rotating, and translating etc [6].
		More generally, the concept of subspace [14] can also be used to derive the so-called kernel PCA method [2] based on the replacement of dot product with an appropriate kernel function [1].
	www.cfar.umd.edu /~wyzhao/RS/Research_Statement.html (2004 words)

	PlanetMath: linear manifold
		A linear manifold is, in other words, a linear subspace that has possibly been shifted away from the origin.
		examples of linear manifolds are points, lines (which are hyperplanes), and
		This is version 3 of linear manifold, born on 2003-11-26, modified 2005-10-29.
	planetmath.org /encyclopedia/LinearManifold.html (113 words)

	Pattern Recognition : SVM decision boundary based discriminative subspace induction
		PCA, ICA and LDA are typical linear dimension reduction techniques used in the pattern recognition community, which simultaneously generate a set of nested subspaces of all possible dimensions.
		This section serves two purposes: (1) to formulate the concept of sufficient subspace for classification in rigorous mathematical form, and (2) to reveal the potential parallelism between classification and regression on the common problem of sufficient dimension reduction.
		For the same subspace, the time cost of NN is reduced from 40.2 to 4.06 s with a 15% decrease in error.
	www.cs.cmu.edu /~zhangjy/pr05/pr05-paper.html (4357 words)

	Subspaces of the Vector Space
		of the subspaces is a subspace of the vector space
		The set of all possible linear combination of the set Z is called the span of the set
		is a subspace, the arbitrary linear combination of the elements os the set Z belongs to the subspace
	www.cs.ut.ee /~toomas_l/linalg/lin1/node6.html (323 words)

	Linear Algebra project
		If V is a linear space with scalars in a field K acting on the left (suppose that K is not commutative), then the dual V* is a right K-linear space (but not a left K-linear space).
		A (linear) subspace has a basis and you want to express each vector of the subspace as a linear combination of the basis.
		By linear ambient I mean a linear space that's the mother of all subspaces in a given problem.
	mate.dm.uba.ar /~caniglia/Linear/index.html (1503 words)

	Linear span - Wikipedia, the free encyclopedia
		In the mathematical subfield of linear algebra, the linear span, also called the linear hull, of a set of vectors in a vector space is the intersection of all subspaces containing that set.
		The linear span of a set of vectors is therefore a vector space.
		Theorem 1: The subspace spanned by a non-empty subset S of a vector space V is the set of all linear combinations of vectors in S.
	en.wikipedia.org /wiki/Linear_span (412 words)

	Math21b, Fall 2003, Linear Algebra and Differential Equations
		Question: Is there any difference between a space and a linear space and between a linear space and a subspace, and between a space and a subspace.
		If a linear space is a subset of an other linear space, one calls it a "linear subspace".
		Since any linear space is also a subspace of itself, it is also a "linear subspace".
	www.math.harvard.edu /archive/21b_fall_03/faq.html (400 words)

	Subspace Identification for Linear Systems (Site not responding. Last check: 2007-10-10)
		The subspace identification theory is linked to the theory of frequency weighted model reduction, which leads to new implementations and insights.
		The implementation of subspace identification algorithms is discussed in term of the robust and computationally efficient RQ and singular value decompositions, which are well-established algorithms from numerical linear algebra.
		The applicability of subspace identification algorithms in industry is further illustrated with the application of the Matlab files to ten practical problems.
	homes.esat.kuleuven.be /~vanovers/bookann.html (292 words)

	Nonlinear Compression Techniques (Site not responding. Last check: 2007-10-10)
		Two layer networks perform a projection of the data onto a linear subspace.
		A helix is 1-D, however, it does not line on a 1-D linear subspace.
		This was reduced to 5 dimensions using linear PCA to obtain the image in the center.
	www.willamette.edu /~gorr/classes/cs449/Unsupervised/nonlinearPCA.html (168 words)

	Robotics Institute: Robust Subspace Clustering by Combined Use of kNND Metric and SVD Algorithm
		The major issue in subspace clustering is to obtain the most appropriate subspace from the given noisy data.
		The remaining data provide a good initial inlier data set that resides in a linear subspace whose rank (dimension) is upper-bounded.
		Such linear subspace constraint can then be exploited by simple algorithms, such as iterative SVD algorithm, to (1) detect the remaining outliers that violate the correlation structure enforced by the low rank subspace, and (2) reliably compute the subspace.
	www.ri.cmu.edu /pubs/pub_4743.html (350 words)

	in theory: Polynomials and subspaces
		, of dimension n/c, and a linear function L, such that f and L have agreement 1/2 + eps' when restricted to V. Here c=c(k) is a constant that depends only on k and eps'=eps'(eps,k) is a constant that depends only on eps and on k.
		If f has agreement 1/2+eps with a linear function L in a subspace V of dimension n-t, then there is a linear function L' such that f and L' agree on a fraction at least 1/2+eps/2
		Its dimension-3 Gowers uniformity is 1, but, if you could find a subspace V of dimension n-o(n) on which f agrees with a linear function on a 1/2+eps' fraction of inputs, then you could also find a linear function that agrees with f on a 1/2+eps/2
	in-theory.blogspot.com /2006/06/polynomials-and-subspaces.html (630 words)

	DI/FCUL \| Informações sobre o DI
		We describe a new hierarchical linear subspace indexing method will based on the generic multimedia indexing (GEMINI) approach, which does not suffer from the dimensionality problem.
		The approach will be demonstrated on image indexing, in which the subspaces correspond to different resolutions of the images.
		In the next subspace additional metric information corresponding to a higher resolution is used to reduce this set.
	www.di.fc.ul.pt /sobre/?abstract&report_ref=2006-03 (202 words)

	ch32
		Be able to find the Nullspace of a matrix A. Understand the concept of linear combinations of vectors and spanning sets.
		Linear combinations and spans are important enough to rate their own definition.
		can be written as a linear combination of the vectors in the given set.
	isolatium.uhh.hawaii.edu /linear/ch3/ch32.htm (462 words)

	S.O.S. Mathematics CyberBoard :: View topic - Vector Spaces Q's
		Which of the following sets are linear subspaces of the space M of n x n matrices.
		f(V)) is a linear subspace of W. b) Show that the kernel of f (i.e.
		) is a linear subspace of V. (1) 0 belongs to
	www.sosmath.com /CBB/viewtopic.php?t=26406 (699 words)

	Univ at Albany: Math: W. F. Hammond: Math 424/524
		In class: second approach to the basis of a linear map with respect to a pair of bases; the effect of change of bases on the matrix of a linear map.
		In class: linear independence of a subset of a vector space; basis of a vector space as a maximal linearly independent set; Zorn's lemma; existence of a basis for any vector space modulo Zorn's lemma.
		In class: the linear map given by a matrix; the subspace spanned by a finite sequence; linear independence of a finite sequence; the subspace spanned by a subset of a vector space.
	math.albany.edu:8000 /math/pers/hammond/course/mat424524f2002/assgt (1073 words)

	some subspace questions
		It would be helpful for you to write out your attempt at showing one of them is, or isn't, a subspace, so we can see where you have problems.
		It is impossible to prove that A invertible and B invertible implies A+B is invertible, isn't it?
		basically inverse images of linear functions and linear operations, are flat like subspaces, and inverse images under more complicated operations are not.
	www.physicsforums.com /showthread.php?t=120008 (861 words)

	Math Forum - Ask Dr. Math Archives: College Linear Algebra
		If w1 and w2 are subspaces of the vector space V, how can I prove that their intersection is also a subspace of V? Intersection Point of Two Lines [07/22/2003]
		Under the operations of matrix addition and multiplication, prove that this is a subspace of M22.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /library/drmath/sets/college_linearalg.html?s_keyid=16260685&f_keyid=16260687&start_at=41&num_to_see=40 (909 words)

	Math Forum Discussions - Re: looking for an example - invariant subspace
		Re: looking for an example - invariant subspace
		> V c R^n is a linear subspace.
		> [A-invariant subspace W, w element W => A_w element of W] > Then = V + AV +....
	www.mathforum.org /kb/thread.jspa?forumID=13&threadID=1476146&messageID=5286292 (188 words)

	Non-linear Point Distribution Models
		PCA projects the data into a linear subspace with a minimum loss of information by multiplying the data by the eigenvectors of the covariance matrix (S).
		This is akin to the polygonal representation of a surface.
		However by projecting the training set down into the linear subspace as derived from PCA, the dimensionality and therefore computation complexity of the non-linear analysis can be reduced significantly to facilitate statistical and probabilistic analysis of the training set.
	www.ee.surrey.ac.uk /Personal/R.Bowden/publications/cvonline/nlpdm/index.html (2364 words)

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