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	Linear independence - Wikipedia, the free encyclopedia
		In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection.
		Equivalently, a family is dependent if a member is in the linear span of the rest of the family, i.e., a member is a linear combination of the rest of the family.
		The concept of linear independence is important because a set of vectors which is linearly independent and spans some vector space, forms a basis for that vector space.
	en.wikipedia.org /wiki/Linear_independence (488 words)

	PlanetMath: linear independence
		Otherwise, if this conditions fails, the vectors are said to be linearly dependent.
		Furthermore, an infinite set of vectors is linearly independent if all finite subsets are linearly independent.
		As an alternate characterization of dependence, we have that a set of of vectors is linearly dependent if and only if some vector in the set lies in the linear span of the other vectors in the set.
	planetmath.org /encyclopedia/LinearIndependence.html (143 words)

	RESPONSE Statement
		For each dependent variable, the response functions are a set of linearly independent cumulative logits, obtained by taking the logarithms of the ratios of two probabilities.
		For each dependent variable, the response functions are a set of linearly independent generalized logits, obtained by taking the logarithms of the ratios of two probabilities.
		For each dependent variable, the response functions are a set of linearly independent marginals, obtained by deleting the marginal probability corresponding to the last level.
	www.sfu.ca /sasdoc/sashtml/stat/chap22/sect20.htm (1686 words)

	Learn more about Vector (spatial) in the online encyclopedia. (Site not responding. Last check: 2007-10-24)
		Which vector is the correct one depends upon the orientation of the vector space, i.e.
		Because the cross product depends on the choice of coordinate systems, its result is referred to as a pseudovector.
		Second, the scalar triple product is zero if and only if the three vectors are linearly dependent, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane.
	www.onlineencyclopedia.org /v/ve/vector__spatial_.html (1961 words)

	The Need for Deflation
		, however, the occurrence of a first linearly dependent vector in (4.27) does not mean that the block Krylov sequence is exhausted.
		Of course, in finite precision arithmetic, it is impossible to distinguish between exactly linearly dependent and almost linearly dependent vectors.
		Therefore, in practice, almost linearly dependent vectors also have to be detected and deleted.
	www.cs.utk.edu /~dongarra/etemplates/node132.html (300 words)

	Exam 1 Study Guide Answers
		Thus, the columns of the matrix are linearly dependent.
		Also, for the vectors to be linearly dependent, the system would need a free variable.
		(a) The columns of B are linearly dependent regardless of the number of pivots.
	www.sci.wsu.edu /math/faculty/martin/Math220/exams/sg1soln.html (1262 words)

	sciforums.com - Vector Spaces
		It seems to me that a set of vectors IS linearly dependent iff they can be expressed as a linear combination of the other vectors in that set, and obviously independant if it cannot.
		a linearly independent vector doesn t have to be perpendicular to this line, but it must at least have some component that is perpendicular.
		Okay, I get that linearly independent means 'geometrically' that for R3 the set of two vectors do not lie on the same line or in the same plane.
	www.sciforums.com /showthread.php?t=27327 (3255 words)

	5 (Site not responding. Last check: 2007-10-24)
		Linearly dependent iff at least one of the vectors in S is expressible as a linear combination of the other vectors in S.
		Linearly independent iff no vector in S is expressible as a linear combination of the other vectors in S.
		A set with exactly two vectors is linearly independent iff neither vector is a scalar multiple of the other.
	www.apsu.edu /vandergriffj/spring99/3450/503.html (158 words)

	No Title
		Two vectors are linearly dependent if and only if one vector is a scalar multiple of the other.
		Vectors are linearly dependent if and only if any one of the vectors is a linear combination of the others.
		Hence the given vectors are not linearly dependent.
	www.csm.uwe.ac.uk /~rlaister/Educs1w8/Educs1w8.html (231 words)

	AMCA: Locally linearly dependent operators by Peter Semrl (Site not responding. Last check: 2007-10-24)
		We also study countable families of locally linearly dependent bounded operators acting on Banach spaces.
		M. Bresar and P. Semrl, On locally linearly dependent operators and derivations, Trans.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/b/w/06.htm (214 words)

	[No title]
		The column vectors of a 5x4 matrix must be linearly dependent.
		If two nonzero vectors are linearly dependent, then each of them is a scalar multiple of the other.
		If vectors U, V, W are linearly dependent, then vector W must be a linear combination of U and V. False.
	orion.math.iastate.edu /hentzel/class.307.ICN/February_26.ppt (2900 words)

	Homework 3 Solutions (Site not responding. Last check: 2007-10-24)
		#2 Linearly dependent since the second vector is (-5/3) times the third vector plus (-19/6) times the first vector.
		(b) The set is linearly dependent for all values of h.
		In this case, this span is a line, OR (c) {T(u),T(v)} are linearly independent, in which case the span is a plane.
	www.evergreen.loyola.edu /~mpknapp/teaching/235-s01/HW3.html (494 words)

	Bases of Vector Spaces, the Basis Problem
		If we want uniqueness, the spanning set must also be linearly independent.
		is linearly dependent, then one of the vectors can be written as a linear combination of the others.
		is linearly dependent, one of the vectors in
	distance-ed.math.tamu.edu /Math640/chapter3/node11.html (1642 words)

	r2 (Site not responding. Last check: 2007-10-24)
		are linearly dependent is equivalent to the condition that at least one of the functions is a multiple of the other.
		If there is a non-trivial solution, then you don't know whether they are linearly dependent or independent.
		There are three cases depending on the nature of the roots of this quadratic equation.
	www.math.okstate.edu /~myersr/2233/review/r2 (1504 words)

	Linear Dependence of Vectors. Basis of the Vector Space.
		Prove that if the column-vectors of determinant are linearly dependent, then the determinant equals 0.
		is called a maximal linearly independent subset if V is linearly independent and it is not a proper subset of any linearly independent subset of the set U.
		is a basis of the space iff it is the maximal linearly independent subset.
	www.cs.ut.ee /~toomas_l/linalg/lin1/node7.html (377 words)

	Murad S. Taqqu -- Articles
		``The asymptotic dependence structure of the linear fractional Lévy motion'' (with A. Astrauskas and Joshua B. Levy).
		``Testing for long-range dependence in the presence of shifting means or a slowly declining trend, using a variance-type estimator'' (with Vadim Teverovsky).
		``Estimating long-range dependence in the presence of periodicity: an empirical study'' (with Alberto Montanari and Vadim Teverovsky).
	math.bu.edu /people/murad/articles.html (3443 words)

	3-3 (Site not responding. Last check: 2007-10-24)
		If there is at least one point at which it is not zero, then the functions are linearly independent.
		Unfortunately this doesn't tell us that the functions are linearly dependent.
		This says that the functions are linearly independent on this interval.
	www.math.okstate.edu /~myersr/2233/homework/3-3 (279 words)

	REVIEW TEST 1 - MAT242 (Site not responding. Last check: 2007-10-24)
		Also remember that any set of more than n vectors in R^n is linearly dependent so, for example, in problem 4 on pag.
		Three vectors are linearly dependent if they are coplanar.
		Also remember that if a set of vectors is linearly dependent it is possible to write one of the vectors in the set as a linear combination of the others; practice on this.
	math.la.asu.edu /~tracogna/MAT242/review2.html (436 words)

	Linearly Independent Sets of Vectors
		If the set is not linearly independent, it is called linearly dependent.
		Linear dependence of a set of two or more vectors means that at least one of the vectors in the set can be written as a linear combination of the others.
		, depending on the degree of dependence of the three vectors.
	distance-ed.math.tamu.edu /Math640/chapter3/node9.html (877 words)

	The Gramian and K-Volume in N-Space: Some Classical Results in Linear Algebra (Site not responding. Last check: 2007-10-24)
		We give a formula for determining when a set of k vectors in n-space is linearly independent, and if so, what is the volume of the parallelepiped with these vectors as its sides.
		and equals zero iff the vectors are linearly dependent.
		If they are not linearly independent, instead take W to be some k-dimensional subspace containing them.
	www.jyi.org /volumes/volume2/issue1/articles/barth.html (1227 words)

	Deflation
		As in the case of the Hermitian band Lanczos method, the use of multiple starting vectors necessitates a suitable deflation procedure to delete linearly and almost linearly dependent vectors in the block Krylov sequences.
		, in the sequence (7.60) is linearly dependent on vectors to the left of
		Similarly, an exact deflation in the left block Krylov sequence (7.61) means that a vector in the sequence (7.61) is linearly dependent on previous Krylov vectors in (7.61) and that this vector and all its
	www.cs.utk.edu /~dongarra/etemplates/node257.html (283 words)

	[No title]
		The dependence of these vectors in R^N is checked by computing all determinants det(pi(vj)) where {v1,...,vk} runs over the various subsets of {1,...,N} of cardinality k.
		Again, there may be some savings in time depending on whether expansion or evaluation is quicker, but the total time needed for a definitive (rather than probabilisitic) answer is in either case going to depend mightily on the size of that matrix.
		The problem, of course, is that of establishing dependence and the corresponding linear relations, and I agree that, once again, we need the "special nature" of the polynomials in question.
	www.math.niu.edu /~rusin/known-math/95/polynom (6731 words)

	l_solns_3 (Site not responding. Last check: 2007-10-24)
		Proof #1: If are linear dependent, one is a linear combination of the others so we can delete it without changing the span.
		Since of the ones you have saved, each is not in the span of the preceding, the ones saved are linearly independent, by a lemma we have discussed.
		Since the original list was a maximal linearly independent set, the new list, being larger, must be linearly dependent.
	www.math.ucla.edu /~baker/115ah.1.01f/handouts/l_solns_3/node16.html (755 words)

	Differential Equations (Math 3401) - Second Order DE's - More on the Wronskian
		This fact is used to quickly identify linearly independent functions and functions that are liable to be linearly dependent.
		So, this means that two linearly dependent functions can be written in such a way that one is nothing more than a constant time the other. Go back and look at both of the sets of linearly dependent functions that we wrote down and you will see that this is true for both of them.
		In other words, if the functions are linearly dependent then we can write at least one of them in terms of the other functions.
	tutorial.math.lamar.edu /AllBrowsers/3401/Wronskian.asp (1443 words)

	20. (Site not responding. Last check: 2007-10-24)
		However, the essential aspect of linear dependence is that you are not dealing with this trivial condition, but that at least one of the numbers
		are linearly dependent, because 2A + B - C = 0.
		In any case, there exists no base consisting of more than n vectors; as we have just seen, more than n vectors are always linearly dependent, while the vectors of a base must be linearly independent.
	kr.cs.ait.ac.th /~radok/math/mat3/m120.htm (932 words)

	problemset-june07
		The first and second columns are linearly dependent and no 1 column is linearly dependent, so the minimum weight of
		columns from block I cannot be linearly dependent.
		In this case, there is a linearly dependent set of
	www.math.unl.edu /~bbockelm/problemset-june07 (752 words)

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