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Topic: Dimension linear algebra

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	MTH 211: Linear Algebra
		Linear algebra is a place where all these questions can be addressed.
		Linear models are often the first step towards understanding more complicated models.
		This course begins where your high school algebra and geometry courses probably ended: solving systems of linear equations and understanding the geometry of 2 and 3 dimensional spaces.
	math.smith.edu /Local/guide/node27.html (182 words)

	PlanetMath: linear algebra
		Linear algebra is the branch of mathematics devoted to the theory of linear structure.
		From the geometric point of view, “linear” is synonymous with “straight”, and consequently linear algebra can be regarded as the branch of mathematics dealing with lines and planes, as well as with transformations of space that preserve “straightness”, e.g.
		Linearity is a very basic notion, and consequently linear algebra has applications in numerous areas of mathematics, science, and engineering.
	planetmath.org /encyclopedia/LinearAlgebra.html (481 words)

	PlanetMath: dimension
		The word dimension in mathematics has many definitions, but all of them are trying to quantify our intuition that, for example, a sheet of paper has somehow one less dimension than a stack of papers.
		One common way to define dimension is through some notion of a number of independent quantities needed to describe an element of an object.
		This is version 7 of dimension, born on 2003-10-18, modified 2004-03-29.
	planetmath.org /encyclopedia/Dimension3.html (510 words)

	Linear Algebra for Informatics
		The original lecture notes have benefited from extant notes on linear algebra by John Meldrum and on polynomials by Andrew Ranicki.
		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)

	Linear algebra
		Linear algebra is the branch of mathematics concerned with the study of vectorss, vector spaces (or linear spaces), linear transformations, and systems of linear equations.
		Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis.
		Linear operators 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.knowledgefun.com /book/l/li/linear_algebra_1.html (797 words)

	Linear transformation - Wikipedia, the free encyclopedia
		In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
		In the language of abstract algebra, a linear transformation is a homomorphism of vector spaces.
		Differentiation is a linear transformation from the space of all differentiable functions to the space of all functions.
	en.wikipedia.org /wiki/Linear_transformation (1103 words)

	Linear algebra
		Linear algebra is the branch of mathematics concerned with the study of vectorss, vector spaces (or linear spaces), linear transformations, and systems of linear equations.
		Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis.
		Linear operators 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.sciencedaily.com /encyclopedia/linear_algebra_1 (858 words)

	Glossary - Linear Algebra
		The dimension of the column space is called the rank of the matrix, and is equal to the dimension of the column space.
		The rank of a matrix is the dimension of the row space, which is equal to the dimension of the column space.
		The dimension of the row space is called the rank of the matrix, and is equal to the dimension of the column space.
	www.math.umbc.edu /~campbell/Math221/Glossary (1250 words)

	Dimension -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-12)
		Dimension (from (Any dialect of the language of ancient Rome) Latin "measured out") is, in essence, the number of (Click link for more info and facts about degrees of freedom) degrees of freedom available for movement in a space.
		In the physical sciences and in engineering, the dimension of a physical quantity is the expression of the class of (Click link for more info and facts about physical unit) physical unit that such a quantity is measured against.
		The Hausdorff dimension is defined for all (A set of points such that for every pair of points there is a nonnegative real number called their distance that is symmetric and satisfies the triangle inequality) metric spaces and, unlike the Hamel dimension, can also attain non-integer real values.
	www.absoluteastronomy.com /encyclopedia/d/di/dimension.htm (1201 words)

	Dimension
		In mathematics, dimensions are the parameters required to describe the position and relevant characteristics of any object within a conceptual space —where the dimensions of a space are the total number of different parameters used for all possible objects considered in the model.
		The inductive dimension of a topological space may refer to the small inductive dimension or the large inductive dimension, and is based on the analogy that n+1-dimensional balls have n dimensional boundaries, permitting an inductive definition based on the dimension of the boundaries of open sets.
		The Krull dimension of a commutative ring, named after Wolfgang Krull (1899 - 1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.
	articles.gourt.com /en/Dimension (1660 words)

	Basis (linear algebra) - Wikipedia, the free encyclopedia
		In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space.
		The latter result is known as the dimension theorem.
		The dimension of R[x] is therefore equal to aleph-0.
	en.wikipedia.org /wiki/Basis_(linear_algebra) (1033 words)

	MTH-1A24 : Linear Algebra II
		A linear transformation is determined by its effect on a basis, and, given bases, we can encode all the information needed to describe a linear transformation in a matrix.
		The origins of linear algebra are in the middle of the 18th century when mathematicians began to analyse systems of (linear) equations more closely.
		Linear Transformations: Definition and examples (the linear transformation arising from a matrix; examples in analysis; geometric examples such as rotation and projection).
	www.mth.uea.ac.uk /maths/syllabuses/0001/1A2401.html (585 words)

	Egwald Mathematics - Linear Algebra: Matrices and Matrix Decomposition
		Vectors and their operations are described on the linear algebra: vectors web page.
		Polynomials and their roots are described on the linear algebra: polynomials web page.
		Systems of linear equations and their solutions are described on the linear algebra: linear equations web page.
	www.egwald.com /linearalgebra/matrices.php (3984 words)

	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 geometric multiplicity of an eigenvalue c of a matrix A is the dimension of the eigenspace of c.
		The nullity of a linear transformation is the dimension of its null space.
	www.caam.rice.edu /~zhang/caam310/glossary.html (1005 words)

	Section I. Geometric Algebra
		Improvements in the formulation of linear algebra are suggested to simplify its intimate relation to projective geometry.
		Relations among Clifford algebras of different dimensions are interpreted geometrically as "projective and conformal splits." The conformal split is employed to simplify and elucidate the pin and spin representations of the conformal group for arbitrary dimension and signature.
		For example, the Jordan form for a linear transformation is shown to be equivalent to a canonical factorization of the unit pseudoscalar.
	modelingnts.la.asu.edu /html/GeoAlg.html (717 words)

	Algebra
		College Algebra Demystified by Rhonda Huettenmueller (McGraw-Hill Professional) One of the most valuable tools acquired in a university education, college algebra is essential for courses from the sciences to computing, engineering to mathematics.
		Algebraic Theory of Automata and Languages by Masami Ito (World Scientific Publishing Company) Although there are some books dealing with algebraic theory of automata, their contents consist mainly of Krohn–Rhodes theory and related topics.
		Linear Algebra: A Geometric Approach by Ted Shifrin, Malcolm Ritchie Adams (W.H. Freeman) We begin Chapter 1 with a treatment of vectors, first in 1<82 and then in higher dimensions, emphasizing the interplay between algebra and geometry.
	www.wordtrade.com /science/mathematics/algebra.htm (7408 words)

	Linear Algebra Review
		Linear Algebra has become as basic and as applicable as calculus, and fortunately it is easier.
		In linear algebra, by convention, the column vector is viewed as the basic case.
		The normal linear algebra convention is that vectors are symbolized with lower-case constants/variables (a, b, c,...
	www.ling.upenn.edu /courses/ling525/linear_algebra_review.html (5609 words)

	Linear-Algebra
		Linear algebra is the study of linear sets of equations and their transformation properties.
		In addition to being used to describe the study of linear sets of equations, the term "linear algebra" is also used to describe a particular
		In particular, a linear algebra L over a field F has the structure of a ring with all the usual axioms for an inner addition
	adwan.net /Linear-Algebra.html (300 words)

	Linear Algebra Review
		Linear Algebra has become as basic and as applicable as calculus, and fortunately it is easier.
		In linear algebra, by convention, the column vector is viewed as the basic case.
		The normal linear algebra convention is that vectors are symbolized with lower-case constants/variables (a, b, c,...
	ling.upenn.edu /courses/Spring_2003/ling525/linear_algebra_review.html (5601 words)

	Northeastern University, Department of Mathematics
		Linear algebra is introduced and matrix methods are used to solve larger systems of differential equations.
		Linear functions and their matrices are introduced, the derivative is defined as a linear function (Jacobian), and the general chain rule is expressed in matrix form.
		Starting with the algebra and geometry of complex numbers, basic derivative and contour integral properties are developed for elementary algebraic and transcendental functions as well as for other analytic functions and functions with isolated singularities.
	www.math.neu.edu /undergrad/ugcatalog.html (5966 words)

	Rank (linear algebra) - Wikipedia, the free encyclopedia
		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.
		then the rank of A can also be defined as the dimension of the image of f, or as n minus the dimension of the kernel of f (see linear map for a discussion of image and kernel).
		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.wikipedia.org /wiki/Rank_of_a_matrix (730 words)

	The Origins of Modern Algebra
		But rather this was algebra in more or less the sense we use the word today (but without thinking of it in abstract terms), namely the study of structures in which one could work in very much the same way that traditional algebra operates in the realm of rational numbers, real numbers, or complex numbers.
		Some of these structures were: the complex numbers, the quaternions, various algebraic number rings (certain subrings of the complex numbers), in addition to the algebra of matrices developed by Sylvester and Cayley and the algebra of logic developed by Boole.
		During this time, the development of Lie groups and algebras (which are non-associative) was proceeding and some of the fundamental concepts in the theory of associative algebras (the concept of the radical, for instance) were developed first for Lie algebras.
	www.math.hawaii.edu /~lee/algebra/history.html (3349 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 (376 words)

	Linear Algebra worksheets for Maple R6 (Site not responding. Last check: 2007-10-12)
		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)

	Linear Algebra - Mathematics, Algebra, Linear Algebra,
		Course 18.06: Linear Algebra (Spring 2006) Department of Mathematics Massachusetts Institute of Technology General Information Lecturer: Alan Edelman, room 2-343, e-mail edelman AT math Lectures: MWF 11 room 54-100 Course Administrator: Alice Chan, room 2-588, phone 3-4110, e-mail alicec Textbook: Introduction to Linear Algebra, 3rd Edition by Gilbert Strang published by Wellesley-Cambridge Press.
		Elements of Abstract and Linear Algebra Edwin H. Connell This is a foundational textbook on abstract algebra with emphasis on linear algebra.
		Linear Algebra Diagnostic Test Instructions: This test is intended as a self-test to determine if your linear algebra skills are sufficient for this course.
	www.studysphere.com /Site/Sphere_2071.html (1025 words)

	Linear Algebra
		Some of them are obscure; mayhap even original with me. This exposition also provides documentation for the library of Linear Algebra programs that I am writing (from scratch) concurrently in the C-language, which recently became accessible to me.
		They already were solving simultaneous linear algebraic equations before written history.
		In April 1980, I was retained to write a library of Linear Algebra routines, to be written in a proprietary dialect of Fortran IV, to run on the Data General Mainframe computer.
	www.rism.com /LinAlg/linear_algebra.htm (1243 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 (1122 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 dimension of one of the term-document matrices examined was 100,000 (number of terms or words) by 2,559,430 (number of documents or home pages), which contained 421 million nonzero entries.
	crd.lbl.gov /~osni (1789 words)

	Linear Algebra With Applications Chapter 10 -- True or False (Site not responding. Last check: 2007-10-12)
		The number of vectors in a basis is called the dimension.
		The dimension of a subspace V is denoted dim(V).
		The kernel and image of a linear transformation are subspaces of the domain and codomain, respectively.
	cwx.prenhall.com /bookbind/pubbooks/bretscher/chapter10/truefalse1/deluxe-content.html (112 words)

	Hamel dimension : Dimension (linear algebra) (Site not responding. Last check: 2007-10-12)
		The dimension of a vector space V is the cardinality (i.e.
		The Hamel dimension is a natural generalization of the dimension of Euclidean space, since E
		However, the Hamel dimension depends on the base field, so while R has dimension 1 when considered as a vector space over itself, it has dimension c (the cardinality of the continuum) when considered as a vector space over Q (the rationals).
	www.eurofreehost.com /di/Dimension_(linear_algebra).html (267 words)

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