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Topic: Rank (linear algebra)

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	PlanetMath: rank of a linear mapping
		Composition of linear mappings does not increase rank.
		"rank of a linear mapping" is owned by yark.
		This is version 10 of rank of a linear mapping, born on 2002-02-19, modified 2007-01-18.
	planetmath.org /encyclopedia/Rank8.html (0 words)

	Linear algebra Encyclopedia
		Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations.
		Linear algebra also has a concrete representation in analytic geometry and it is generalized in operator theory.
		Linear maps take elements from a linear space to another (or to itself), in a manner that is compatible with the addition and scalar multiplication given on the vector space(s).
	www.hallencyclopedia.com /topic/Linear_algebra.html (1371 words)

	Matrix Theory and Linear Algebra - MSN Encarta
		Matrix Theory and Linear Algebra, interconnected branches of mathematics that serve as fundamental tools in pure and applied mathematics and are becoming increasingly important in the physical, biological, and social sciences.
		A matrix is a rectangular array of numbers or elements of a ring (see Algebra).
		If A is a matrix of rank r, then at least one set of r row, or column, vectors is a linearly independent set, and every set of more than r row, or column, vectors is a linearly dependent set.
	encarta.msn.com /encyclopedia_761565107/Matrix_Theory_and_Linear_Algebra.html (0 words)

	math lessons - Rank (linear algebra)
		In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent.
		The rank of a matrix plus the nullity of the matrix equals the number of columns of the matrix (this is the "rank theorem" or the "rank-nullity theorem").
		In those generalisations, column rank, row rank, dimension of column space and dimension of row space of a matrix may be different from the others or may not exist.
	www.mathdaily.com /lessons/Rank_(linear_algebra) (621 words)

	Linear Algebra.
		Linear algebra is a branch of algebra which deals with linear vector spaces, linear operators as well as linear, bilinear and quadratic functions.
		The results of linear algebra have found application in such diverse fields as optics, quantum mechanics, display addressing, electric circuits, cryptography, computer graphics, economics, linear programming, solution of systems of differential equations, etc. The manipulation of matrices and determinants plays a central role in all applications of linear algebra.
		The concept of the linear vector space took shape towards the end of the nineteenth century and has dominated developments in the field of linear algebra during the twentieth century.
	www.ias.ac.in /currsci/jul25/articles27.htm (1180 words)

	Edwin H. Connell: Elements of Abstract and Linear Algebra
		Linear Algebra: Modules, homomorphisms, quotient modules, products and coproducts, summands, independence, generating sets, free basis, uniqueness of dimension, change of basis, rank of a matrix, geometric interpretation of determinant, nilpotent homomorphisms, eigenvalues, characteristic roots, inner product spaces, orthogonal groups, diagonalization of symmetric matrices by orthogonal matrices (40 pages).
		The goal is to do the minimum amount of abstract algebra necessary to do the linear algebra, and to have material so basic that it is beneficial to students in computer science and the physical sciences.
		Teaching abstract algebra and linear algebra as separate courses results in a loss of synergy and a loss of momentum.
	www.math.miami.edu /~ec/book/author.html (0 words)

	Math 310 - Glossary of Linear Algebra Terms
		The algebraic multiplicity of an eigenvalue c of a matrix A is the number of times the factor (t-c) occurs in the characteristic polynomial of A.
		The rank of a matrix A is the number of nonzero rows in the reduced row echelon form of A; i.e., the dimension of the row space of A.
		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 (0 words)

	Centrality of Linear Algebra
		The separate linear algebra course became a standard part of the college mathematics curriculum in the United States in the 1950s and 60s and some colleges and universities were still adding the course in the early 1970s.
		Many colleges and universities are introducing a second course in linear algebra because they recognize the importance of the subject and the inadequacy of their first course in meeting the diverse needs of the students.
		Linear algebra provides a course that is full of ideas, with material that is rewarding to learn and to teach, and is a subject where both student and teacher can be challenged to their best performance.
	www.maa.org /features/cowen.html (3727 words)

	Student understanding of topics in linear algebra
		He also cites that textbooks in linear algebra are written based on the assumptions "that students recognize models and solve problems by translating them to isomorphic but abstract structures; and that they can apply the principles of abstract setting to solve problems.
		Since course content is dependent on the needs of a number of disciplines for a linear algebra course, the math department must maintain a constant liaison with these departments in order to monitor their needs in higher level mathematics.
		After taking the linear algebra course, students often come away knowing how to perform certain algorithms but they have not acquired the intuition relating knowledge of the mathematics to selection of the method for analysis, design, and control of physical systems (Wang 237).
	www.physics.umd.edu /rgroups/ripe/perg/plinks/linalg.htm (1312 words)

	Linear Algebra Examples - Wolfram Mathematica
		Least Squares QR When A is full rank, a more accurate way to solve the problem is to use the QR decomposition as follows.
		is differentiable and differentiation is a linear operation.
		The matrix represents the linear operator when the grid is flattened into a vector.
	reference.wolfram.com /mathematica/tutorial/LinearAlgebraExamples.html (0 words)

	MA106 Linear Algebra
		The module contains a theoretical algebraic core, whose main idea is that of a vector space and of a linear map from one vector space to another.
		It discusses the concepts of a basis in a vector space, the dimension of a vector space, the image and kernel of a linear map, the rank and nullity of a linear map, and the representation of a linear map by means of a matrix.
		They must also understand the equivalence of linear maps between vector spaces and matrices and be able to row reduce a matrix, compute its rank and solve systems of linear equations.
	www.maths.warwick.ac.uk /pydc/blue/blue-MA106.html (0 words)

	Linear Algebra worksheets for Maple R6 (Site not responding. Last check: 2007-10-16)
		One of the sub-project is to teach linear algebra in a computer classroom,using the computer whenever it seems appropriate for the material to be taught.
		Exploring Linear Algebra with Maple, sponsored by the MAA in the summer of 2005.
		Just Enough Maple for Linear Algebra - This is a preliminary worksheet designed to introduce the students to the mechanics of working with Maple worksheets.
	euler.slu.edu /courseware/LinearAlgebra-R95/LinalgMaster.html (1664 words)

	Homework Tutoring in Math, Statistics, Physics, Chemistry, Programming, Economics, Electronics, Psychology, English, ...
		College Algebra Overiew: Linear and quadratic equations and inequalities; graphs of equations, including lines, circles, parabolas; composition, inverses of functions; transformations of graphs; linear and quadratic models; polynomials; exponentials, logarithms; counting; probability.
		Overview: Separable and first-order linear equations with applications, 2nd order linear equations with constant coefficients, method of undetermined coefficients, simple harmonic motion, 2x2 and 3x3 systems of linear ODE's with constant coefficients, solution by eigenvalue/eigenvectors, nonhomogenous linear systems.
		Linear and quadratic equations and inequalities; graphs of equations, including lines, circles, parabolas; composition, inverses of functions; transformations of graphs; linear and quadratic models; equations and inequalities involving polynomials and rational functions; exponentials and logarithms, with applications.
	www.homeworktutoring.com /online_courses.shtml (427 words)

	Numerical Linear Algebra
		For example, Google uses it to rank the most important web pages, a structural engineer uses it to measure the maximum load of a bridge, a sound engineer uses it to measure the lowest resonating frequency in a concert hall.
		The SVD has many applications: multiple linear regression, factor analysis, computer graphics, face recognition, noise reduction, information retrieval, robotics, gene expression analysis, computational tomography, geophysical inversion (seismology), image compression, image deblurring, face recognition, using optics linear sensitivity matrices to analyze spacecraft dynamics, visualization of chemical databases, and latent semantic indexing (LSI).
		The Hilbert matrix is ill-conditioned, and most linear algebra packages have difficulty inverting this matrix for sufficiently large N. Note that although it inverts smaller matrices without reporting an error, there is substantial error in the results.
	www.cs.princeton.edu /introcs/95linear (0 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.
		At the far right are several large areas of activity in numerical linear algebra and related topics, typically, the study of individual matrices or transformations between (large-dimensional) real vector spaces.
	www.math.niu.edu /~rusin/known-math/index/15-XX.html (0 words)

	Osni Marques' Home Page
		In this study, a linear model for the Earth structure is obtained by discretizing the Earth into layers, and the layers into cells.
		The velocity of wave propagation in each cell is represented by a set of parameters that account for anisotropy, location (and correction) of sources and receivers, etc. These parameters are written in matrix form and the goal is to fit the model by means of some known data.
		The goal was to study conformational changes of proteins by means of linear combinations of normal modes coordinates.
	crd.lbl.gov /~osni (0 words)

	Linear Algebra I
		Defining linear transformations in terms of the images of the elements in a basis of a linear space.
		The linear space of linear transformations between a pair of linear spaces.
		Linear independence of eigenvectors and the existence of a full set of eigenvectors if all eigenvalues are distinct.
	www.cs.nyu.edu /cs/faculty/widlund/courses/linI05/index.htm (1018 words)

	ERCIM meeting at KVL April 2005
		Linear algebra and data analysis are driving forces behind these developments and this workshop will focus on some of the key aspects, applications and new developments.
		Linear algebra provides a natural setting for many algorithms in computational statistics, due to the availability of a variety of efficient and robust algorithms for matrix computations, such as QR factorizations, SVD etc.
		On the other hand, the multilinear rank of an order-k tensor is a k-tuple of integers but it has the advantage of preserving most, if not all, common features of the matrix rank.
	www.models.kvl.dk /ercim (0 words)

	Linear Algebra
		It is easy to handle the rank 1 case (to within a natural equivalence) they consist of row vectors or column vectors.
		Spaces of linear transformations of equal rank, Linear and Multilinear Algebra 13 (1983), 231-239 (with R. Westwick).
		Extensions to the Kronecker-Weierstrass theory of pencils, Linear and Multilinear Algebra 29 (1991), 235-241.
	www.cs.otago.ac.nz /staffpriv/mike/HomePages/linearalgebra.html (0 words)

	CliffsNotes::Linear Algebra: The Rank Plus Nullity Theorem
		Also, the rank of this matrix, which is the number of nonzero rows in its echelon form, is 3.
		The sum of the nullity and the rank, 2 + 3, is equal to the number of columns of the matrix.
		Since the nullity is the difference between the number of columns of A and the rank of A, the nullity of this matrix is 6 − 2 = 4.
	www.cliffsnotes.com /WileyCDA/CliffsReviewTopic/The-Rank-Plus-Nullity-Theorem.topicArticleId-20807,articleId-20794.html (0 words)

	Linear Algebra I: Syllabus
		Definition of bases; reduction of a spanning set and extension of a linearly independent set to a basis; proof that all bases have the same size.
		Linear transformations from one (real) vector space to another.
		The matrix representation of a linear transformation with respect to fixed bases; change of basis and co-ordinate systems.
	web.comlab.ox.ac.uk /oucl/courses/topics03-04/LAI/syllabus.html (136 words)

	NCSU Faculty Publications Repository: NCSU Libraries (Site not responding. Last check: 2007-10-16)
		Linear algebra and its applications, 379(2004 Mar 1), 85-112.
		Linear algebra and its applications, 366(2003 Jun 1), 157-172.
		Linear algebra and its applications, 272(1998 Mar. 15), 155-168.
	www.lib.ncsu.edu /repository/search.php?page=journal&pos=1&jid=419 (866 words)

	Egwald Mathematics - Linear Algebra: Systems of Linear Equations
		Consequently, b will not be represented exactly as a linear combination of the columns of A. I discuss least squares methods, also called multiple regression methods, on the Statistical web pages.
		Since the matrix S has 3 positive singular values, the rank of the matrix A is 3.
		If the rank of A, rank(A), equals n, its columns and rows are linearly independent and the problem has a unique solution.
	www.egwald.ca /linearalgebra/linearequations.php (0 words)

	Math 311 Linear Algebra - Syllabus
		The course provides an introduction to the concepts and theories that form the foundation of Linear Algebra.
		Linear operators; compositions; kernels and images; dimensional theorem for linear operators between finite-dimensional spaces.
		Students should be able to use a Computer Algebra System to perform Linear Algebra calculations, explore basic concepts and create lab worksheets.
	isolatium.uhh.hawaii.edu /linear/syllabus.htm (172 words)

	VII. REVIEW OF LINEAR ALGEBRA*
		A set of m vectors which span a space V of dimension m are said to form a basis for that space.
		The row rank of A is the largest number of linearly independent rows.
		Column rank is the largest number of linearly independent columns.
	courses.temple.edu /economics/notes/algebra/NBUCK1.htm (0 words)

	Teaching Linear Algebra
		"Matrix Algebra in Economics," written by Clopper Almon is quite enthusiastic about applications of linear algebra to statistics (least square models and related topics), to modeling an economy, to the maximization of functions of many variables subject to constraints and to dynamical systems defined by difference equations.
		Gerald Porter in "Writing About Linear Algebra: Report on an Experiment" talks about a linear algebra course for non-mathematics majors in which he had the students write a ten page chapter to supplement the text material on subspaces, spanning sets, basis and dimension, lines, planes and hyperplanes.
		Gerald Porter in "Linear Algebra and Affine Planar Transformations" gives concrete ideas which are used in computer graphics and which are easily accessible to the linear algebra student.
	www.maa.org /reviews/linearalg.html (1787 words)

	Linear Algebra, MTH 513
		We will cover concepts of linear algebra that will be useful in a wide variety other courses and applications.
		A basic knowledge of linear algebra such as is covered in an undergraduate course is required as background for this course.
		In many mathematical models of such complex situations, the transformations involved turn out to be linear in the sense that the sum of two inputs is transformed into the sum of their individual outputs and a multiple of an input is transformed into that multiple of the original output.
	hypatia.math.uri.edu /~eaton/mth513.f01.htm (1120 words)

	Linear Algebra - Preliminary lectures
		Linear Algebra consists mostly of studying matrix calculus.
		Linear Algebra allows for formalising and solving many typical problems in different engineering topics.
		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 (0 words)

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