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PlanetMath: linearly independent In the case of two vectors, linear dependence means that one of the vectors is a scalar multiple of the other. As an alternate characterization of dependence, we also have the following. Cross-references: linear combination, characterization, scalar multiple, finite, subset, infinite, vectors, scalars, field, vector space planetmath.org /encyclopedia/LinearIndependence.html   (132 words)

Linear Algebra for Informatics They are a revised version of the ones used in the 2004-2005 session, which were themselves revised due to changes in the syllabus from the ones used in the 2003-2004 session. Linear algebra is the study of vector spaces and linear maps. We will discuss subspaces, linear (in)dependence, bases, dimension, linear maps and linear transformations and their relation to matrices, the effect of changing basis, eigenvalues and eigenvectors and diagonalisation. xmlearning.maths.ed.ac.uk   (171 words)

VIAS Encyclopedia: Linear Dependence   (Site not responding. Last check: 2007-10-15) Linear independence is important for many aspects of data analysis. A general rule is that a set of n vectors of order m shows linear dependence if n is greater than m. Linear independence is closely related to the rank of a matrix. www.vias.org /encyclopedia/cc_matrix_lindepend.html   (140 words)

Linear Algebra - Preliminary lectures 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. It is unfortunately often the case that it is not possible to calculate eigenvalues and eigenvectors explicitly. vision.unige.ch /~marchand/teaching/linalg   (1819 words)

PlanetMath: dependence relation This is equivalent to the definition of linear dependence. This is equivalent to the definition of algebraic dependence. This is version 4 of dependence relation, born on 2004-04-21, modified 2005-08-05. www.planetmath.org /encyclopedia/DependenceRelation.html   (87 words)

Linear independence, bases and dimension  -  Linear (in)dependence In the plane, any three or more vectors form a linearly dependent set, whereas any set consisting of one nonzero vector or any set consisting of two non-collinear vectors is linearly independent. four or more vectors are linearly dependent, whereas any two non-collinear vectors or any three non-coplanar vectors are linearly independent. Show that a set consisting of two vectors is linearly dependent if and only if one vector is a scalar multiple of the other. xmlearning.maths.ed.ac.uk /lecture_notes/linear_independence_bases_dimension/linear_independence/linear_independence.php   (109 words)

Nat' Academies Press, Health Effects of Exposure to Low Levels of Ionizing Radiations: Time for Reassessment? (1998) A linear-quadratic dependence on dose is used that has the form, where Dγ is the gamma ray organ dose and Dn the neutron organ dose. THE DOSE DEPENDENCE OF RBE The analysis of potential curvilinearity is, in the case of the atomic-bomb survivor data, inseparable from the issue of accounting for the RBE of neutrons. Dependence of the gamma ray and the neutron colon dose on distance from hypocenter in Hiroshima according to DS86 (solid lines) and neutron dose tentatively modified according to Straume and others (1992) (broken line). www.nap.edu /books/0309061768/html/33.html   (3467 words)

PlanetMath: quadratic sieve The quadratic sieve method of factoring depends upon being able to create a set of numbers whose factorization can be expressed as a product of pre-chosen primes. The linear dependence among them is used to find a combination of the vectors which sum up to the zero vector in We would be wasting time and space to employ other primes in our factoring and linear combinations. www.planetmath.org /encyclopedia/QuadraticSieve.html   (382 words)

Math 310 - Glossary of Linear Algebra Terms The nullity of a linear transformation is the dimension of its null space. A linear transformation T from V to W is orthogonal if T(v) has the same length as v for all vectors v in V. The rank of a linear transformation (and hence of any matrix regarded as a linear transformation) is the dimension of its range. www.math.uic.edu /math310/glossary.html   (1060 words)

Conclusion   (Site not responding. Last check: 2007-10-15) This notion is further supported by the obseravtion of a linear term in the single gaussian fits, which provide a crude estimate of the second moment. The weakening of the linear term at 1.5T (or conversely, the strengthening of the linear term at 0.5T) was surprising indeed. One possible explanation for the observed field dependence is quasiparticle scattering off of the vortex cores, which we know to be static, as evidenced by the field-shifted results of Fig. musr.physics.ubc.ca /theses/Sonier/MSc/node41.html   (703 words)

Linear Dependence of the Postsunset Equatorial Anomaly Electron Density on Solar Flux and Its Relation to the Maximum ... Linear Dependence of the Postsunset Equatorial Anomaly Electron Density on Solar Flux and Its Relation to the Maximum Prereversal E x B Drift Velocity Through Its Dependence on Solar Flux The relation of E x B drift, which is also a linear function of Sa, is determined using measurements of Nemax vs. Sa measured at Bogota in the anomaly plotted as a function of E x B vs. Sa measured at Jicamarca at the dip equator. The result is that Nemax is a linear function of E x B, which is in agreement with that found previously during solar maximum. www.stormingmedia.us /79/7948/A794824.html   (492 words)

Correlation The correlation is 1 in the case of an increasing linear relationship, −1 in the case of a decreasing linear relationship, and some value in between in all other cases, indicating the degree of linear dependence between the variables. If the variables are independent then the correlation is 0, but the converse is not true because the correlation coefficient detects only linear dependencies between two variables. To get a measure for more general dependencies in the data (also nonlinear) it is better to use the correlation ratio which is able to detect almost any functional dependency, or mutual information which detects even more general dependencies. www.sciencedaily.com /encyclopedia/correlation   (439 words)

Linear Dependence, Column Space, Null Space, and Bases   (Site not responding. Last check: 2007-10-15) In this lab we will talk about some of the most important concepts in linear algebra: the column and null space of a matrix, linear dependence, and bases of vector spaces. Every time you had to solve a system of homogeneous, linear equations, you were actually looking for an element in the null space of the corresponding coefficient matrix. Likewise a problem of solving a non-homogeneous system of linear equations Ax = b, is a problem of determining whether b lies in the column space of A. math.ucsd.edu /~math20f/Fall/Lab5F/Lab5F.html   (1287 words)

quadratic sieve   (Site not responding. Last check: 2007-10-15) Once enough vectors are collected to form a set which contains a linear dependence, this linear dependence is exploited to find two squares which are equivalent modulo n. If there are k prime numbers in the factor base, then each number which can be factored within the factor base is stored as a k-dimensional vector where the i-th component of the vector for y gives the exponent of the i-th prime from the factor base in the factorization of y. The linear dependence among them is used to find a combination of the vectors which sum up to the zero vector in... www.citidel.org /dlxml/oai_dc-oai:PlanetMath:QuadraticSieve.xml   (549 words)

\bigrmbf TEMPERATURE DEPENDENCE OF IN-PLANE RESISTIVITY OF YBCO:   (Site not responding. Last check: 2007-10-15) We investigated the origin of variation in the magnitude and temperature dependence of the normal state resistivity frequently observed in different YBCO single crystals (or YBCO thin films) of the same Tc. The magnitude of resistivity associated with a superlinear dependence is lower than that with a linear dependence. The superlinear temperature dependence of resistivity was thought to be related to a "pseudo-gap" (" spin-gap"). phyvax.ir.miami.edu:8001 /hts04/prog_abstracts/jung_jan_1.html   (209 words)

Exercise 3.11   (Site not responding. Last check: 2007-10-15) We assume as an induction hypothesis that any k (k-1)-vectors are linearly dependent, which means that there must be a linear dependence among any k (k-1)-component vectors. This means if we can take our k+1 k-vectors and produce k linear distinct linear combinations of them that are each (k-1)-vectors, we can use the induction hypothesis to give us a linear dependence among these which will produce a linear dependence among our original vectors. Any non-zero coefficient in the linear combination of linear combinations will give rise to a non-zero coefficient of the corresponding one of our original vectors and we are really done. ocw.mit.edu /ans7870/18/18.013a/textbook/HTML/chapter03/exercise11.html   (296 words)

Linear Dependence of Vectors. Basis of the Vector Space. Prove that each set of vectors that contains the null vector is linearly dependent. Prove that if the column-vectors of determinant are linearly dependent, then the determinant equals 0. is not expressable as a linear combination of vectors of V. www.cs.ut.ee /~toomas_l/linalg/lin1/node7.html   (377 words)

3.5 Linear Dependence and Independence   (Site not responding. Last check: 2007-10-15) A linear dependency among vectors v(1) to v(k) is an equation, A set of vectors is said to be linearly independent if there is no linear dependence among them, and linearly dependent if there is one or more linear dependence. Then v(1), v(2) and v(3) are linearly dependent because there is the relation www-math.mit.edu /18.013A/HTML/chapter03/section05.html   (123 words)

Correlation: Pitfalls and Alternatives Increasingly, however, correlation is being used as a dependence measure in general risk management, often in areas where the assumption of multivariate normal risks is completely untenable - such as credit risk. In using correlation as an all-purpose dependence measure and transferring CAPM thinking to general risk management, many integrated risk management systems are being built on shaky foundations. The recognition that correlation is often an satisfactory measure of dependence in financial risk management is not in itself new. www.defaultrisk.com /pp_corr_08.htm   (478 words)

Composition Dependence of GMR in Fe/Ag/CoxFe1-x Sandwiches   (Site not responding. Last check: 2007-10-15) The magnitude of the giant or spin-valve magnetoresistance in ferromagnet/paramagnet/ferromagnet sandwiches depends on the choice of ferromagnet. In this study, the dependence of the giant magnetoresistance on the composition of the alloy layer in a series of Fe/Ag/Co x The anisotropic magnetoresistance of these sandwiches shows a simple linear dependence on x, the Co fraction, but the giant magnetoresistance has a non-monotonic composition dependence. www.wsrcc.com /alison/jmmm   (126 words)

Linear dependence of the postsunset equatorial anomaly electron density on solar flux and its relation to the maximum ... Linear dependence of the postsunset equatorial anomaly electron density on solar flux and its relation to the maximum prereversal The relation to E × B drift, which is also a linear function of Sa, is determined using measurements of Nemax versus Sa measured at Bogota in the anomaly plotted as a function of E × B versus Sa measured at Jicamarca at the dip equator. The result is that Nemax is a linear function of E × B, which is in agreement with that found previously during solar maximum. www.agu.org /pubs/crossref/2004/2004JA010528.shtml   (404 words)

linearly dependent? - Information Technology Services   (Site not responding. Last check: 2007-10-15) Of course, on can show that if the functions involved all satisfy the same linear, homogeneous differential equation, THEN their Wronskian is either always 0 or never 0. Normally, Linear Dependence for an arbitrary differentiable set of Functions is defined on an OPEN INTERVAL "I" and requires the Wronskian to be zero (0) everywhere on "I". Being (0) at 1 point in "I" (or a finite number of points in "I") does not usually indicate Linear Dependence if there exists at least 1 other point in "I" for which the Wronskian is NON-zero. www.physicsforums.com /archive/t-64008_linearly_dependent?.html   (351 words)

15: Linear and multilinear algebra; matrix theory Linear algebra, sometimes disguised as matrix theory, considers sets and functions which preserve linear structure. Classic topics in linear algebra and matrix theory are at the center of the diagram: 15A03: Vector spaces, 15A04: Linear transformations, 15A15: Determinants, and 15A21: Canonical forms (e.g. Also crowded near the center of the diagram are several fields concerned with linear spaces and linear transformations, and in some cases the reflection of those ideas in the corresponding matrices. www.math.niu.edu /~rusin/known-math/index/15-XX.html   (1605 words)

[No title] Prior to the move the gain had a linear za dependence all the way down to za=0. The linear dependence was probably being caused by the focus error in the rails (green lines). The only question with cband is that the cband focus curve (red line bottom plot) shows that its current focus is close to 1256.35 feet. www.naic.edu /%7Ephil/callbn/focchangeaug02.html   (687 words)

Danville Area Community College - Catalog This course is a study of introductory linear algebra.  Basic techniques are introduced involving vectors and matrices; vector spaces and subspaces; linear dependence, independence, and transformations and dimension; determinants; and orthogonality. This course is a basis for a first undergraduate course in linear algebra. The student shall be evaluated on the basis of quizzes, homework, major examinations, final examination, and progress during the course.  At least 60% accuracy is necessary to pass the course. www.dacc.cc.il.us /courses/index.php?course=MATH137&print=yes   (146 words)

Recombination of photoexcitations in conjugated polymers: dispersive processes   (Site not responding. Last check: 2007-10-15) The photoinduced absorption spectra contain two broad bands, whose relative intensities strongly depend on the protonation state of the polymers [1] and whose modulation frequency dependence is distinctly different. The value of the fractional exponent is sample dependent and is a quantitative measure for the distribution of the relaxation centers [2]. The sub-linear dependence on the pump intensity at low modulation frequencies and the super-linear dependence at high modulation frequencies both indicate a bi-molecular recombination mechanism. www.esqsec.unibe.ch /pub_94.htm   (288 words)

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