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Topic: Nullspace

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Linear subspace

Empty set

	Null space
		The null space (also spelt nullspace) of a m by n matrix A is the set of all vectors v which are solutions to the equation:
		This is the same as the kernel of A if it is interpreted as a linear map.
		The rank of any matrix plus its nullity equals the number of columns of that matrix.
	www.ebroadcast.com.au /lookup/encyclopedia/nu/Nullspace.html (82 words)

	Matrix Invariants
		Given an m x n sparse matrix A over a ring R, return the nullspace of A (or the kernel of A, considered as a linear transformation or map), which is the R-space consisting of all vectors v of length m such that v.A = 0.
		Given an m x n sparse matrix A over a ring R, return a (dense-representation) basis matrix of the nullspace of A. This is a matrix N having m columns and the maximal number of independent rows subject to the condition that N.A = 0.
		This function has the advantage that the nullspace is not returned as a R-space, so echelonization of the resulting nullspace may be avoided.
	www.umich.edu /~gpcc/scs/magma/text764.htm (435 words)

	The Nullspace of Electrocardiography (Site not responding. Last check: 2007-10-13)
		The matrix A has a non-vanishing nullspace: different vectors x can be found, that belong to this nullspace and lead to potentials b smaller than noise.
		Note that the higher the index, the more the basisvector belongs to the nullspace (the smaller are the resulting potentials).
		Generator distributions with large components of the nullspace are difficult and different regularization techniques will have different spatial filtering characteristics that are interesting to investigate.
	www-ibt.etec.uni-karlsruhe.de /forschung/nullspace/nullspace_en.htm (1165 words)

	Blair Lee, Stephen Lee, Alfredo Vincenty #2
		In the HELP menu we looked up the colspace and nullspace functions and had to use them to solve the next set of problems on pages 58 and 75 of Demko.
		For 4f we got column space with dimension three and no nullspace (although for some reason we got a set of empty brackets instead of a zero matrix).
		For problem 3a, b, and c on page 75 we had to find the range and nullspace of the given matrices.
	coweb.math.gatech.edu:8888 /model/938 (471 words)

	MUG: Bug in nullspace, Maple V.5 (9.11.98)
		Nullspace appears to work for square matrices, but not for non-square ones.
		The columns of V corresponding to the zero singular values (i.e., the last n columns of V where n is the nullity of A, n=# of columns of A - rank(A), will be a basis for Nul A. An orthonormal basis, in fact.
		The nullspace of a square matrix (for example) contains only the nullvector, unless the matrix can be reduce by Gaussian elimination to a matrix with a row/rows of zeros.
	www.math.rwth-aachen.de /mapleAnswers/html/653.html (1129 words)

	Guaranteed Optimization: Proving Nullspace Properties of Compilers - Veldhuizen, Lumsdaine (ResearchIndex)
		By analogy with linear algebra, we define the nullspace of an optimizer as those programs it reduces to the empty...
		we prove the nullspace property for a model optimizer of an imperative language.
		By analogy with linear algebra, we define the nullspace of an optimizer as those programs it reduces to the empty program.
	citeseer.ist.psu.edu /528872.html (734 words)

	nullspace
		The nullspace of non-zero 4x4 matrix cannot contain a set of 4 lin.
		Does the same hold for dim of nullspace (nullity): it has to have at least one solution (trivial, where everyting = 0, but that does not mean that the nullity is an empty set!) ?
		In the case that the nullspace is the zero vector (and only in this case), the nullity of the matrix is 0, because (as I mentioned in my last post), the dimension of the vector space containing only the zero vector is zero (a basis for the vector space has zero elements - ie.
	www.physicsforums.com /showthread.php?t=73335 (660 words)

	Interactive Redundant Robotics: Control of the Inverted Pendulum with Nullspace Motion - Schreiber, Ott, Hirzinger ...
		At the example of an inverted pendulum the "interactive nullspace motion " is introduced, where the user is able to modify the configuration as a subtask while balancing the pendulum as primary task.
		Di#erent ways of nullspace interaction are shown, either contact-free by a...
		Interactive redundant robotics: Control of the inverted pendulum with nullspace motion.
	citeseer.ist.psu.edu /554552.html (307 words)

	Charles Whittington & John Bard
		For the problems in Exercise 3, we were asked to find the columnspace/range, and the nullspace of the system of equations using colspace and nullspace.
		We used the colspace and nullspace commands to find the range/columnspace and the nullspace, respectively.
		The nullspace was the line that goes through the vector [-1,1].
	coweb.math.gatech.edu:8888 /model/886 (408 words)

	Witt's theorem - Wikipedia, the free encyclopedia
		A maximal nullspace is a nullspace which is not contained in any larger nullspace.
		Corollary Let (E, b) be a symmetric, non-degenerate bilinear form, then every maximal nullspace has the same dimension, which is the maximum dimension of any nullspace.
		Then, because, both spaces are nullspaces, this map of vector spaces is automatically metric.
	en.wikipedia.org /wiki/Witt's_theorem (426 words)

	Nullspaces and Solutions of Systems
		Given an m x n matrix A over a ring R, return the nullspace of A (or the kernel of A, considered as a linear transformation or map), which is the R-space consisting of all vectors v of length m such that v.A = 0.
		Given an m x n matrix A over a ring R, return a basis matrix of the nullspace of A. This is a matrix N having m columns and the maximal number of independent rows subject to the condition that N.A = 0.
		The nullspace N of A so that adding any elements of N to any rows of V will yield other solutions to the system.
	www.umich.edu /~gpcc/scs/magma/text748.htm (668 words)

	[No title]
		For n x m matrices A: (Definitions) The nullspace of A is the set of solutions to Ax=0 (x and 0 are mx1 vectors) The columnspace of A is the set of all linear combinations of its columns; it is a subspace (hyperplane) in (n.
		The rowspace of A is the set of all linear combinations of its rows; it is a subspace (hyperplane) in (m.
		In fact, the rowspace and the nullspace are orthogonal complements: the rowspace contains every vector that is perpendicular to a vector in the nullspace and vice versa.
	www.mathcs.richmond.edu /~khoke/323/hw1andlinearalgegra.doc (487 words)

	Citebase - Computing the Rank and a Small Nullspace Basis of a Polynomial Matrix (Site not responding. Last check: 2007-10-13)
		Computing the Rank and a Small Nullspace Basis of a Polynomial Matrix
		We reduce the problem of computing the rank and a nullspace basis of a univariate polynomial matrix to polynomial matrix multiplication.
		For an input n x n matrix of degree d over a field K we give a rank and nullspace algorithm using about the same number of operations as for multiplying two matrices of dimension n and degree d.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:cs/0505030 (232 words)

	nullspace
		I try to post things that I find interesting as a means to keep in touch with college friends, but those are few and far between when you enter the doldrum of work, sleep, work, sleep, play etc.
		Nullspace has been around for four years, and it will never have any sort of focus since it never had it to begin with.
		nullspace is powered by WordPress 2.0.5 and K2 Beta Two r160 by Michael and Chris
	blogs.belldandy.us /nullspace (2866 words)

	OOPS - algebra-12
		Then all the potential differences are 0 and that x is in the nullspace.
		As this formula, the DIMENSION OF NULLSPACE OF trans(A) is m – r, OK This is the NUMBER OF LOOPS, number of independent loops.
		Because that [1, 1, 1, 1] vector in the nullspace the columns were not independent.
	oops.editme.com /algebra-12?show-menu=false (4197 words)

	Left Nullspace of A
		The Left Nullspace of A is just the nullspace of Transpose[A].
		If M is the GJSF of A, then M=D.A Partition D by rows into D=Transpose[D11 D21], where D11 corresponds to the nonzero rows of M and D21 corresponds to the zero rows of M. The columns of Transpose[D21] form the standard basis for the Left Nullspace of A. In[52]:=
		The last three rows correspond to the nonzero rows of M. The standard basis for the Left Nullspace of A is
	www.ma.iup.edu /projects/CalcDEMma/linalg2/linalg217.html (227 words)

	GeoSci 236: The Fundamental Theorem of Linear Algebra
		, because the nullspace component is mapped to zero, by construction, thus not changing the end result in any way.
		In other words, it will have a nonempty nullspace, a situation also explicitly addressed by our schematic.
		This may be justified as the least conjectural, or most parsimonious, solution (given our imperfect information), but it is far from clear that these are the solution's most desirable attributes.
	geosci.uchicago.edu /~gidon/geosci236/fundam (618 words)

	ue:oh core content / nullspace
		If the multiverse is everything, NullSpace is nothing.
		Scientists theorize that since we have various realities in the multiverse and these realities tend to have fixed sizes beyond which stuff gets very fuzzy and unsafe, that means there must be a NullSpace, or absence of reality.
		Since it's impossible to prove the existence of something that's the absence of something except by inverse, NullSpace becomes more of a cultural construct after you get beyond the basic definition.
	www.pixelscapes.com /unrealestate/openhouse/core/nullspace.shtml (154 words)

	Math Forum Discussions
		>Given an mxn matrix A and its Nullspace N(A) which is (n-m)xn matrix.
		The nullspace of a matrix is not a matrix.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /kb/thread.jspa?messageID=3830540&tstart=0 (120 words)

	Faith - Page 1559 - Advent Children.forums (Site not responding. Last check: 2007-10-13)
		Last edited by nullspace : Dec 13 2006 at 06:57 PM.
		Even without a bra, her boobs defy gravity on both the game and AC.
		Last edited by nullspace : Dec 14 2006 at 03:36 AM.
	forums.adventchildren.net /showthread.php?p=2657085 (1589 words)

	The Linear Algebra Package ``linalg'' (Site not responding. Last check: 2007-10-13)
		returns a basis for the nullspace of the matrix
		During the calculation the nullspace of a matrix has to be computed.
		If this is not possible, the empty list will be returned (this can happen for example if the matrix is defined over
	math.berkeley.edu /~mgu/mupad/mupad_html_help/linalg53.html (88 words)

	unreal estate / encyclopedia / nullspace
		Information in this encyclopedia is subject to change without warning as the story develops.
		Of course, nobody's ever seen NullSpace, since by definition it is that which does not exist.
		"I'll put your ass in NullSpace if you don't have the money by friday," for instance.
	www.pixelscapes.com /unrealestate/encyc/nullspace.shtml (181 words)

	Echelon Form and Nullspace
		Nullspace (many different techniques including sparse, p-adic and modular algorithms)
		(The nullity is nearly always 1, and the integer entries of the non-zero null vector usually have about 450 digits each.) This can be compared with an algorithm of J. Buchmann and D. Squirrel for computing nullspaces in the Lidia system[
		Even assuming conservatively that they have used a 200Mhz processor (so we halve their time), Magma is thus about 750 (sic) times faster.
	magma.maths.usyd.edu.au /magma/Features/node151.html (115 words)

	Example 2: Rank, nullity, nullspace, basis for a nullspace
		Example 2: Rank, nullity, nullspace, basis for a nullspace
		Find the rank, the nullity, the nullspace and a basis for the nullspace of the matrix
		=> A basis for the nullspace: { (-2, 1, 0, 0), (-17/5, 0, 3/5, 1) }
	www.ee.oulu.fi /~mpa/matreng/eem3_2-2.htm (191 words)

	Transparencies for Lecture 17 (Site not responding. Last check: 2007-10-13)
		Definition of the Nullspace (Kernel) of a Matrix
		Example: Computing the Nullspace of a 2x2 Matrix
		How to Compute a Basis for the Nullspace of a Matrix
	www.ncsu.edu:8010 /Project25/MA305/Spring98/17.html (85 words)

	Nullspace: The fancy cars...the women and the caviar
		Nullspace: The fancy cars...the women and the caviar
		What exactly did I buy? Come by the house in a couple weeks and I'll play some for you.
		Don't have a Windows Live ID? Sign up now.
	rogerku.spaces.live.com /Blog/cns!1pT5M2xlxl35p6-QR7JZNNjw!287.entry (190 words)

	nullspace's Profile – Users at Last.fm
		Type in some bands or artists you like, separated by commas.
		Want to know how similar your music taste is to nullspace’s?
		It’s easy, create your own musical profile and share your taste with your friends.
	www.last.fm /user/nullspace (196 words)

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