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Topic: Linear combination

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Linearly independent

Linear subspace

Vector space

Linear transformation

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	Linear span - Wikipedia, the free encyclopedia
		In the mathematical subfield of linear algebra, the linear span of a set of vectors is the set of all linear combinations of the vectors.
		The linear span of a set of vectors is therefore a vector space but unlike a basis the vectors need not be linearly independent.
		The linear span is an example of a set-builder notation.
	en.wikipedia.org /wiki/Linear_span (569 words)

	Linear combination - Wikipedia, the free encyclopedia
		In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics.
		Most of this article deals with linear combinations in the context of a vector space over a field, with some generalisations given at the end of the article.
		Or, if S is a subset of V, we may speak of a linear combination of vectors in S, where both the coefficients and the vectors are unspecified, except that the vectors must belong to the set S (and the coefficients must belong to K).
	en.wikipedia.org /wiki/Linear_combination (1066 words)

	[No title]
		Linear combination: SaiYi Expected value: SaimYi Estimate of mean: Sai`Yi ¡2uÿn.þMÿèZþu$þ þ þ þ þ þ þ þ þ þ þ þ þ þª Â¦y0¾Ú÷ðB ð sð“n“”¾±v¿ÿ ?ð ÿÿÿÀÀÀ@ÿÿ¿ÂAî(ï ù Àð¸`ððVð( ð ðð² ð ÓðNð—¿¿ÿ ?ðÛeðÃ— ð¦ 0ðd¢ ð ãðTøã—¿¿Àÿ?ðSâêS ðà¨Variance of the linear* combination.
		The variance of a linear combination is the sum of the variances of the individual variances (with squared coefficients) plus twice the covariances of the variables (with both coefficients).
		The linear combination reduces to the sum of the variances of the individual variables (with squared coefficients.
	stat.lsu.edu /faculty/geaghan/EXST7005/Spring2000/17LinearCombos.ppt (583 words)

	No Title (Site not responding. Last check: 2007-11-06)
		learn about column vectors and their algebraic properties; understand what a linear combination is; understand what the span of a set of vectors is; learn how to multiply a vector by a matrix; understand the connections between matrix-vector equations and linear systems; understand the algebraic properties of matrix-vector products.
		A linear combination of vectors is a sum of scalar multiples of the vectors.
		A linear system is just a vector equation where the left-hand side of the equation is a linear combination of some vectors with the unknowns for the system used as scalar multipliers.
	www.sci.wsu.edu /math/faculty/genz/220v/lessons/l02/l02.html (304 words)

	Wikinfo \| Linear combination
		can't be zero; in that case, we declare by convention that the result of the linear combination is the zero vector in
		Furthermore, this span is the smallest subspace of
		is indeed a linear combination of the given vectors.
	www.wikinfo.org /wiki.php?title=Linear_combination (1119 words)

	Sampling distribution of a linear combination of means (1 of 4) (Site not responding. Last check: 2007-11-06)
		Further assume that (1) n subjects are sampled randomly from each population with the mean computed for each sample and (2) a linear combination of these means is computed by multiplying each mean by a coefficient and summing the results.
		Let the linear combination be designated by the letter "L." If this sampling procedure were repeated over and over again, a different value of L would be obtained each time.
		The importance of linear combinations of means can be seen in the section "Confidence interval on linear combination of means" where it is shown that many experimental hypotheses can be stated in terms of linear combinations of the mean and that the choice of coefficients determines the hypothesis tested.
	davidmlane.com /hyperstat/A117209.html (222 words)

	Linearly Independent Sets of Vectors (Site not responding. Last check: 2007-11-06)
		Note that the first and last linear combinations yield the same vector (0,0), even though the coefficients are not the same.
		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.
		Linear combinations in complex vector spaces have important applications, as the next examples illustrate.
	distance-ed.math.tamu.edu /Math640/chapter3/node9.html (877 words)

	Linear Combination of Vectors, and the Linear Vector Space.
		Linear Combination of Vectors, and the Linear Vector Space.
		for a set of vectors b and c and some scalars m and n, then a is a linear combination of b and c.
		space is thus a three dimensional linear vector room, where we let each point correspond to a position vector.
	hemsidor.torget.se /users/m/mauritz/math/vect/linear.htm (406 words)

	SparsLinC: Sparse Linear Combination Library
		SparsLinC (Sparse Linear Combination Library) is a library implementing sparse vector linear combinations, optimized for use in the context of automatic differentiation.
		The SparsLinC (Sparse Linear Combination) library provides an implementation of vector linear combinations when p is large and most of the vectors involved in vector linear combination are sparse (that is, for the most part they contain zero entries).
		In the numerical linear algebra literature, the latter two data structures are usually referred to as the ``single-subscript'' and ``compressed subscript'' representation of a sparse vector.
	www-fp.mcs.anl.gov /autodiff/sparslinc.htm (431 words)

	Notes for Exam 3 (Site not responding. Last check: 2007-11-06)
		Note that (3, 4, 2) is not a linear combination of the vectors (1, 1, 0) and (0, 1, 1).
		Thus, (3, 4, 2) is not a linear combination of (1, 1, 0) and (0, 1, 1).
		First, the span of a set of vectors (i.e., all possible linear combinations of those vectors) is actually a subspace, not just a subset of the original vector space.
	www.math.byu.edu /~dsiebert/math343/prevnotes3.html (3141 words)

	Wilmott Forums - why duration retain linear combination relationship? (Site not responding. Last check: 2007-11-06)
		therefore, the convexity of the portfolio is the linear combination of those of the bonds with the investment weights.
		The linear combination relationships of modified duration and convexitie are not so obvious to me, and I don't want to just memorize the conclusion.
		Pp is a linear combination of Pi's (not weighted average, the latter is a special case of the former)
	www.wilmott.com /messageview.cfm?catid=8&threadid=8844 (441 words)

	Lesson 3.3: Writing the gcd(a, b) as an Integral Linear Combination of a and b (Site not responding. Last check: 2007-11-06)
		Thus the gcd 2 is written as an integral linear combination of 22 and 38.
		We need an efficient way to calculate (a, b) as an integral linear combination of a and b, especially when a and b are large integers.
		We continue this process until we have 2 written as an integral linear combination of 22 and 38.
	math.usask.ca /encryption/lessons/lesson03/page3.html (536 words)

	Superposition Principle
		Superposition principle states that a linear combination of solutions to a linear equation is again a solution.
		In general, if V is a linear operator from one vector space to another and we have that Ax = f
		Operator A was assumed to be linear after all.
	www.cut-the-knot.org /do_you_know/superposition.shtml (317 words)

	RGB-DEMO (Site not responding. Last check: 2007-11-06)
		Objective: Near the beginning of a linear algebra course the basic operations of matrix addition and scalar multiplication are combined to define linear combinations of vectors or matrices.
		Prerequisites: Students should be familiar with the basic operations of matrix addition and scalar multiplication and the notion of a linear combination.
		At a single pixel, the color of the pixel is determined by a linear combination of the 3-vectors in
	mathdemos.gcsu.edu /rgb-demo1 (432 words)

	[No title]
		Note that it is not possible to divide a linear combination by another linear combination, as the result is generally not a linear combination.
		The overloaded operators are: '+' for adding two linear combinations; '-' for subtracting two linear combinations; '' for multiplying two linear* combinations; '/' for dividing a linear combination by a number; and '""' for stringifying a linear combination, i.e.
		you can use a linear combination objects in any place where interpolation of variables is possible and get the string representation.
	www.cpan.org /modules/by-module/Text/WIMV/Math-LinearCombination-0.03.readme (811 words)

	Variable-Pixel Linear Combination
		The algorithm, known as Variable-Pixel Linear Reconstruction (or informally as ``drizzling"), preserves photometry and resolution, can weight input images according to the statistical significance of each pixel, and removes the effects of geometric distortion both on image shape and photometry.
		Much of the information lost in sampling can be recovered by combining images which have been shifted by fractions of a pixel between successive exposures.
		The linear weighting scheme employed is statistically optimum when inverse variance maps are used as weights.
	www.cv.nrao.edu /adass/adassVI/hookr.html (1413 words)

	Extension of Euclid's Algorithm
		a and b themselves are expressed as trivial combinations: a = 1·a + 0·b and b = 0·a + 1·b.
		I adopt the usual convention of solving linear equations and omit all the terms in a linear combination but the left-hand side and the two coefficients on the right.
		Moreover, once g is expressed as a linear combination of a and b we know that gcd(a,b)g.
	www.cut-the-knot.org /blue/extension.shtml (487 words)

	Data Interpolation
		Since the coefficients of this linear combination are independent of the tensor indexes, the linear combination of the tensors can be done component-wise.
		We can use trilinear component-wise interpolation because symmetric tensors form a linear subspace in the tensor space: any linear combination of symmetric tensors remains a symmetric tensor, i.e., symmetric tensors are closed under linear combination (the manifold of symmetric tensors is not left).
		On the other hand, component-wise interpolation of eigenvectors and eigenvalues themselves would not lead to correct results, since a linear interpolation between two unit vectors is not a unit vector anymore - the interpolated eigenvector value would leave the manifold of unit vectors.
	www.gg.caltech.edu /~zhukov/research/fiber_tracking/vis02/node5.html (222 words)

	Uniqueness of RREF
		Two matrices of the same size are called row equivalent if every row of the first is a linear combination of the rows of the second and every row of the second is a linear combination of rows of the first.
		In fact, the coefficient of the k th nonzero row R_k of A must be the same as the entry that V has in the j_k spot, where j_k is the number of the column in which R_k has its leading 1.
		Thus, if c_k is the coefficient of R_k in the linear combination, one calculates that the entry of V in the j_k spot is c_k(1) plus a sum of other terms all of which are 0.
	www.math.lsa.umich.edu /~hochster/419/urr.html (671 words)

	solve systems of linear equations by Linear Combination
		Combination of lines means the addition of lines is part of the procedure required to solve the system.
		Through it's simple arithmetic, a sophisticated combination of equations is reduced to a single easy equation.
		Or, these last examples which show how an inconsistent system and a dependent system look when using linear combination to solve the system.
	www.mathnstuff.com /math/algebra/asystem.htm (407 words)

	Linear Algebra - Preliminary lectures (Site not responding. Last check: 2007-11-06)
		Linear Algebra consists mostly of studying matrix calculus.
		It is generally the case that (input or output) data from an experiment is given in a discrete form (discrete measurements).
		Linear Algebra is then useful for solving problems in such a context with applications in topics such as Physics, Fluid Dynamics, Signal Processing and, more generally Numerical Analysis.
	vision.unige.ch /~marchand/teaching/linalg (1819 words)

	R: Fitting Normal Regression Models where Covariance is linear combination of known matrices (Site not responding. Last check: 2007-11-06)
		is a function that fits a linear regression model by REML when the error variance is a linear convex combination of given matrices V1,...
		The identity is included by default as the first matrix in the linear combination unless explicitly excluded.
		Includes the identity matrix as the first of the matrices in the linear combination.
	galton.uchicago.edu /~clifford/Software/reml.html (439 words)

	Vectors in 2-Space (Site not responding. Last check: 2007-11-06)
		When possible the applet shows the linear combination of v1 and v2 necessary to form b.
		Select the "Draw Vectors" button to draw v1, v2, and b in the xy plane, and select the "Draw Linear Combination" button to show the multiples of v1 and v2 whose sum is the vector b.
		The "Explanation" button provides a discussion of the Linear Span and Linear Combination of v1 and v2.
	www.dean.usma.edu /math/research/mathtech/java/LinearAlgebraProject/linearalgebrapage.htm (130 words)

	Linear Combination (Site not responding. Last check: 2007-11-06)
		Compute the linear combination of a matrix and a vector.
		Linear combinations are common in statistics, particularly in linear models and multivariate analysis.
		In Dataplot applications, the LINEAR COMBINATION command is most typically used as an intermediate calculation in a larger macro.
	itl.nist.gov /div898/software/dataplot/refman2/auxillar/linecomb.htm (284 words)

	Mathematical Programming Glossary - L (Site not responding. Last check: 2007-11-06)
		The vector y is a linear combination of vectors x^1, x^2,..., x^m if y = Sum_k{a_k x^k}, where a_k is called the coefficient of x^k.
		When a univariate function is piece-wise linear, it has the form a(i)x + b(i) for x in the interval [c(i), c(i+1)], where a(i) is not equal to a(i+1).
		(The phrase usually means the function is continuous.) This arises in linear programming when considering the optimal value as a function of varying right-hand sides (or cost coefficients) in fixed proportions: b+td (or c+td), where d is an admissible change vector and t is the (scalar) variable.
	carbon.cudenver.edu /~hgreenbe/glossary/L.html (2371 words)

	Linear Combination of Vectors
		A linear combination of vectors is a sum of
		In signal processing, we think of a linear combination as a signal mix.
		Thus, the output of a mixing console may be regarded as a linear combination of the input signal tracks.
	ccrma-www.stanford.edu /~jos/r320/Linear_Combination_Vectors.html (113 words)

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