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Topic: Quotient space (linear algebra)

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	Quotient space - Wikipedia, the free encyclopedia
		In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space.
		The quotient space X/~ is then homeomorphic to Y (with its quotient topology) via the homeomorphism which sends the equivalence class of x to f(x).
		The topological dimension of a quotient space can be more (as well as less) than the dimension of the original space; space-filling curves provide such examples.
	en.wikipedia.org /wiki/Quotient_space (952 words)

	Linear Transformations
		In the previous example, the kernel is the z axis, and the quotient space, or image space, is the xy plane.
		If a linear combination of elements in w becomes 0, then the same linear combination of preimages from y maps to something in the kernel, which is spanned by x.
		The dimension of the space is the sum of the dimensions of the two subspaces, minus the dimension of their intersection.
	www.mathreference.com /la,xform.html (959 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)

	Quotient space (linear algebra) - Wikipedia, the free encyclopedia
		In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero.
		The first isomorphism theorem of linear algebra says that the quotient space V/ker(T) is isomorphic to the image of V in W.
		The quotient space is already endowed with a vector space structure by the construction of the previous section.
	en.wikipedia.org /wiki/Quotient_space_(linear_algebra) (512 words)

	Prof. A. Iarrobino, Math 1302 Linear Algebra II, Class Notes
		The rank of A is the dimension of the row space R(A): it is the number of leading 1's of rows in the row echelon reduced matrix obtained from A by row operations.
		Solution A: (Standard linear algebra method) (assume A is mxn): Row reduce the matrix Ac (A with an extra column c), so that the first n columns are in echelon form.
		The quotient space R^3/N(A) is the one-dimensional vector space of planes P_c: x+2y+3z=c.
	www.math.neu.edu /~iarrobino/AIMath1302.Spr01.classnotes (2482 words)

	Univ at Albany: W. F. Hammond
		A linear map from one vector space to another admits a natural contravariant map from the dual of the second to the dual of the first.
		For finite-dimensional spaces the dual of a short exact sequence is another, and, therefore, the second dual of a short exact sequence is the original.
		The determinant of a finite-dimensional linear endomorphism is characterized as the trace of its highest exterior power; this is equivalent to the statement that the determinant of an n \times n matrix is the unique alternating n-linear form on row space that assigns the value 1 to the identity matrix.
	nyjm.albany.edu:8000 /~hammond/course/linalgcmt.html (1478 words)

	A COURSE IN LINEAR ALGEBRA WITH APPLICATIONS (Site not responding. Last check: 2007-10-16)
		The concept of a quotient space is introduced and related to solutions of linear system of equations, and a simplified treatment of Jordan normal form is given.
		Numerous applications of linear algebra are described, including systems of linear recurrence relations, systems of linear differential equations, Markov processes, and the Method of Least Squares.
		The book is addressed to students who wish to learn linear algebra, as well as to professionals who need to use the methods of the subject in their own fields.
	www.worldscibooks.com /mathematics/6237.html (282 words)

	Linear Algebra by Stephen H. Friedberg [ISBN: 0135371023] - Find Cheap Textbook Prices & Save BIG
		This top-selling, theorem-proof book presents a careful treatment of the principle topics of linear algebra, and illustrates the power of the subject through a variety of applications.
		It emphasizes the symbiotic relationship between linear transformations and matrices, but states theorems in the more general infinite-dimensional case where appropriate.
		I used it for a second course in linear algebra.
	www.gettextbooks.com /isbn_0135371023.html (711 words)

	Algebra
		Algebra is usually presented at three levels: elementary or high school, intermediate or college, and advanced or higher algebra.
		Algebra and geometry were more advanced subjects, involving theory as well as practical matters, and were taught in high school and college.
		The ratio of two quantities is, in algebra, the quotient of their numerical values, written a/b or a:b, where a is called the antecedent, and b the consequent.
	www.du.edu /~jcalvert/math/algebra.htm (17854 words)

	MA 401 Linear Algebra (Site not responding. Last check: 2007-10-16)
		Linear transformations, representation of linear transformations by matrices, rank-nullity theorem, duality and transpose.
		Quotients of groups, Basic examples of groups (including symmetric groups, matrix groups, group of rigid motions of the plane and finite groups of motions).
		Differentiable manifolds (as subspaces of Euclidean spaces), differentiable functions on manifolds, tangent spaces, vector fields, differential forms on manifolds, orientations, integration on manifolds, Stokes' Theorem on manifolds.
	www.math.iitb.ac.in /acad/syllabus.html (1983 words)

	Math 113: Linear algebra and matrix theory (Site not responding. Last check: 2007-10-16)
		the relationship between the algebraic and geometric points of view and matters fundamental to the study and solution of linear equations.
		Topics: linear equations, vector spaces, linear dependence, bases and coordinate systems; linear transformations and matrices; similarity; eigenvectors and eigenvalues; diagonalization.
		Recall our definition of quotient space V/W, where W is a subspace of vector space W. We defined V/W as a set, and claimed that it is in fact a vector space.
	math.stanford.edu /~vakil/113 (1899 words)

	Linear Algebra Glossary (Site not responding. Last check: 2007-10-16)
		Moreover, it is not enough that these vectors span the space; we also require that they be linearly independent, that is, there should be no redundant vectors in the set.
		The columns of an orthogonal matrix are an orthonormal basis for the linear space of vectors of dimension N.
		In such a case, there are no space advantages to be gained by using a special matrix storage format, and so the matrix entries are stored using the standard two dimensional array format provided by the programming language.
	www.csit.fsu.edu /~burkardt/papers/linear_glossary.html (13553 words)

	[No title]
		The projection p: TM -> M exhibits the configuration space* M as a quotient space of the phase space TM. And the zero section s: M -> TM exhibits the configuration space as a subspace of the phase space.
		Here it is: we want Lagrangian subspaces L_x of the tangent spaces T_x(X), these subspaces should vary smoothly with x, and they should fit together in a nice way: they should be "integrable".
		We've got a symplectic vector space V. And Toby muses: >[...] there must be some reason why >(at least when phase space is a vector space) >we can talk about subspaces instead of quotient spaces.
	www.math.niu.edu /~rusin/known-math/00_incoming/symplectic (883 words)

	Quotient Vector Space -- from Wolfram MathWorld
		Algebra > Linear Algebra > General Linear Algebra
		The quotient space is an abstract vector space, not necessarily isomorphic to a subspace of
		, as well as to ensure the quotient space is Hausdorff.
	mathworld.wolfram.com /QuotientVectorSpace.html (82 words)

	Quotient space?
		The quotient space is the set of equivlance classes.
		Obviously for more complicated examples we can't even picture the initial space, never mind using that to construct the quotient space in our heads.
		The concept of quotient spaces comes every now and then in our class.
	www.physicsforums.com /showthread.php?p=353273 (407 words)

	Lecture Outlines - Linear Algebra (Site not responding. Last check: 2007-10-16)
		To any linear programming problem corresponds a dual linear problem: if the original (primal) is a minimization problem, then the dual is a maximization problem and vice versa.
		Lecture Outline: Continued discussing the Rayleigh quotient and showed that its minimum is the smallest eigenvalue of A and its maximum is the largest eigenvalue.
		Motivated eigenvalues and eigenvectors in the context of solving a linear system of differential equations with constant coefficients; defined the characteristic equation and the characteristic polynomial of a matrix which allows us to solve for its eigenvalues.
	www.math.gatech.edu /~demers/linearalgebra/classes.html (3267 words)

	linear
		The conference will be held under the auspices of the Institute of Advanced Studies in Mathematics at the Technion and the International Linear Algebra Society.
		Idempotency of linear combinations of an idempotent matrix with a tripotent one
		The equality of linear transforms of the ordinary least squares estimator and the best linear unbiased estimator
	www.math.technion.ac.il /institute/linear.htm (681 words)

	Advanced Linear Algebra (Applied MS) (Site not responding. Last check: 2007-10-16)
		Prerequisites: The arithmetic of matrices, solving systems of equations using matrix techniques, the matrix of a linear transformation with respect to a given basis, computing determinants and eigenvalues, and using similar matrices to find powers and roots of a given matrix.
		Linear transformations: examples of linear transformations, kernel, image, rank, invertibility, diagonalization, reduced echelon form, matrix of a linear transformation with respect to given bases, similarity, and classical adjoint, and extending a linear transformation defined on a subspace to the vector space by telling what it does to a basis
		Canonical forms: eigenvectors, eigenvalues, characteristic polynomial, minimal polynomial, symmetric matrix, direct sum decomposition, invariant subspaces, and elementary divisors, Jordan and the Rational canonical forms of a matrix, and the Cayley-Hamilton theorem
	www.math.okstate.edu /grad/long-hbk/Advanced_Linear_Algebra_App.html (163 words)

	Linear Algebra-A (Site not responding. Last check: 2007-10-16)
		Fields and vector spaces - basic definitions [doc format]
		Linear maps and their presentation with matrices [doc format]
		Quotient space; tensor product of vector spaces [doc format]
	math.haifa.ac.il /hinich/WEB/lineara.htm (112 words)

	Linear Algebra (Site not responding. Last check: 2007-10-16)
		Cliff uses J with his Linear Algebra classes.
		For example, in the fall of 2003 he used nine J laboratories with his linear algebra class.
		See also At Last for other ideas for linear algebra laboratories.
	ww2.lafayette.edu /~reiterc/j/linalg.html (161 words)

	Prerequisites (Site not responding. Last check: 2007-10-16)
		This comprises all finite fields (one up to isomorphism for each prime power).
		Also, you need to be acquainted with basic linear algebra.
		vector space, linear combination, subspace, basis, standard basis, dimension, linear map, kernel, image, linear subspace, rank, isomorphic, determinant, sum, direct sum, quotient space, dual vector space.
	www.win.tue.nl /~amc/ow/lba/voorkennis.html (111 words)

	Mathematics Archives - Topics in Mathematics - Abstract Algebra
		Algebra: Abstract and Concrete by Frederick M. Goodman
		Lecture Notes, Algebraic sets, Hilbert's Nullstellensatz, varieties over algebraically closed fields, complex analytic manifolds, genus, divisors, linear series, line bundles and the Riemann-Roch theorem.
		Course notes, algebraic numbers and integers, quadratic fields, rings of integers, divisibility and factorization, ideal theory, ideal classes and the class group
	archives.math.utk.edu /topics/abstractAlgebra.html (1342 words)

	4305 Topics (Site not responding. Last check: 2007-10-16)
		2.4 The four fundamental subspaces (column space, nullspace, row space, left nullspace and the "fundamental theorem of linear algebra, part I")
		3.1 Perpendicular vectors and orthogonal subspaces ("the fundamental theorem of linear algebra, part II")
		Last modified: Mon May 9 00:10:38 EDT 2005
	www.math.gatech.edu /~viveros/Linear_Algebra/4305topics.html (115 words)

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