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Topic: Projection (linear algebra)

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	Orthogonal projection - Wikipedia, the free encyclopedia
		In geometry, an orthogonal projection of a k-dimensional object onto a d-dimensional hyperplane (d < k) is obtained by intersections of (k − d)- dimensional hyperplanes drawn through the points of an object orthogonally to the d-hyperplane.
		If such a projection leaves the origin fixed, it is a self-adjoint idempotent linear transformation; its matrix is a symmetric idempotent matrix.
		An orthogonal projection is a bounded operator on a Hilbert space H which is self-adjoint and idempotent.
	en.wikipedia.org /wiki/Orthogonal_projection (298 words)

	PlanetMath: projection
		is called a projection if it acts like the identity on its image.
		Proposition 2 The kernel and image of an orthogonal projection are orthogonal subspaces.
		This is version 5 of projection, born on 2002-07-26, modified 2004-02-25.
	planetmath.org /encyclopedia/Projection.html (170 words)

	Linear Algebra
		Linear Algebra is the study of vector spaces and linear transformations.
		Linear transformations are functions which respect the operations of addition and scalar multiplication.
		Concepts from linear algebra are used in nearly every upper level mathematics course and have become quite important in physics, engineering, and statistics.
	www.iwu.edu /~lstout/LinearAlgebra/F02sylLinearAlgebra.html (973 words)

	MTH-2C13: Linear Algebra II (Site not responding. Last check: 2007-08-12)
		MTH-2C13: Linear Algebra II MTH-2C13: Linear Algebra II Introduction: This course follows on from Linear Algebra I which is a prerequisite.
		Linear transformations are determined by their effect on a basis and so can be described by a matrix.
		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/0405/2C1304.html (361 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)

	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)

	Student understanding of topics in linear algebra
		The increasing demand for student understanding of linear algebra and the growing concern that the present linear algebra course does not adequately meet the needs of the students it serves, prompts further study and curriculum development.
		The first course in linear algebra is a service course for a wide variety of disciplines such as computer science, electrical engineering, other engineering fields such as aerospace engineering and systems engineering, physics, economics, statistics, and operations research.
		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)

	Projection (linear algebra) - Wikipedia, the free encyclopedia
		In linear algebra, a projection is a linear transformation P from a vector space to itself such that P
		An orthogonal projection is a projection for which the range U and the null space V are orthogonal subspaces.
		These projections are also used to represent spatial figures in two-dimensional drawings (see oblique projection), though not as frequently as orthogonal projections.
	en.wikipedia.org /wiki/Projection_(linear_algebra) (822 words)

	MTH-1A24 : Linear Algebra II
		MTH-1A24 : Linear Algebra II MTH-1A24 : Linear Algebra II Introduction: This second semester course follows on from Linear Algebra I which is a prerequisite.
		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.
	www.mth.uea.ac.uk /maths/syllabuses/0203/1A2403.html (584 words)

	Linear Algebra (Site not responding. Last check: 2007-08-12)
		This linearity of the sets of possible solutions is true generally in quantum mechanics, as is the representation of physical variables by operators on the wave functions.
		There can be no inverse operator to a nontrivial projection operator, since the information about components of the vector perpendicular to the projected subspace is lost.
		Since linear operators are represented by matrices, and their action on vectors is standard matrix multiplication, it follows that applying two linear operators in succession is equivalent to matrix multiplication—and, therefore, since matrices do not in general commute, nor do linear operators.
	galileo.phys.virginia.edu /classes/751.mf1i.fall02/751LinearAlgebra.htm (2711 words)

	From Gilbert Strang "Linear Algebra and its Applications", problem 3.2.2
		Linear algebra: projection - Consider two vectors a and b with elements a1,a2,a3 and b1,b2,b3 respectivley.
		Linear algebra problems - I have two questions that I need help with.
		Linear Algebra: Projection matrix question - See page four, the question is question # 2 Show that P^2= P^T = P and that C(P) = C(A) Quadratic Equations and Applications - Assignment 5: Solutions of Quadratic Equations and their Applications 1.
	www.brainmass.com /homework-help/math/other/36269 (267 words)

	Linear Algebra and Linear Programming (Site not responding. Last check: 2007-08-12)
		Comments: The distinction between Elementary Linear Algebra (MATH 230) and MATH 335 is somewhat fuzzy, but the 300-level course places more emphasis on the concept of linear independence, the notion of an abstract vector space, and the theoretical foundations of linear algebra, whereas the 200-level course is more computational and applied.
		The current BSU Linear Algebra course is currently 4 credits, and there may be some objections to reducing it to 3 credits (in which case one alternative would be to give it another number, such as 336).
		Linear Algebra is not a Core course, but it is an important service course for disciplines such as engineering and the physical and computational sciences.
	math.boisestate.edu /~grantham/common_course_draft_report/node16.html (253 words)

	Linear Algebra
		Here you will find information concerning the Linear Algebra (2270-1) class for the Spring 2003 Semester, which meets on Monday, Tuesday, Wednesday, and Friday 8:35AM-9:25AM in LCB 121.
		Also, the second lab project was distributed and is due on Friday 3/14.
		Continued discussion of the image of a linear transformation and notion of kernel.
	www.math.utah.edu /~jfernand/teaching/2270/spring03 (776 words)

	Linear Algebra for Virtual Reality
		This squashing is called projection (that's what movie projectors do, display the world on a 2D surface).
		linear transformation -- I'm describing a change, either in position (by translating), in orientation (by rotating), or in size (by scaling).
		Another type of projection, an orthographic projection, is one in which those railroad tracks never come together and things don't get smaller as they go off into the distance.
	www.mindspring.com /~brownw2/teach.html (1222 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 (1819 words)

	PlanetMath: partial isometry
		Before defining what a partial isometry is, let's recall two familiar concepts in linear algebra: an isometry and the adjoint of a linear map.
		is self-adjoint, this idempotent is in fact a projection.
		For example, if the ring in question is the ring of linear endomorphisms over a Euclidean space (real or complex), then a partial isometry is just a map such that its restriction to the
	planetmath.org /encyclopedia/InitialProjection.html (308 words)

	Linear algebra outline
		A linear map f from V to W is a map (also called function, mapping, or transformation) such that for any points P and Q and scalars s and t,
		Projection onto a line OP parallel to a plane in 3 dimensions, but the interpretation is different: ON is perpendicular to the given plane.
		The most important ideas in connection with abstract vector spaces are linear dependence, independence, and bases.
	www.maths.tcd.ie /~odunlain/1ba1/linalg/outline.html (1020 words)

	Amazon.com: Linear Algebra: Books: Jin Ho Kwak,Sungpyo Hong (Site not responding. Last check: 2007-08-12)
		This is a clean and elegant presentation for a thorough first course in Linear Algebra from the view of pure mathematics.
		Linear algebra continues to be one of the most useful courses in undergraduate mathematics, science, and engineering, and one of the essential tools for industrial scientists.
		The primary aim of this book is to give a clear and rigorous presentation of the basic concepts of linear algebra as a coherent part of mathematics.
	www.amazon.com /Linear-Algebra-Jin-Ho-Kwak/dp/0817639993 (1315 words)

	Orthogonal projection
		What do you think a projection onto a subspace is? To me it is an idempotent linear map.
		a projection is amap onto a subspace, that sends every vector to a vector in the subspace, and leaves vectorsd that are already in the subspace where they are.
		Indeed any linear map f such that f^2 = f is asuch a projection.
	www.physicsforums.com /showthread.php?t=114671 (452 words)

	Math 220 - Linear Algebra by Alex Degtyarev (Site not responding. Last check: 2007-08-12)
		Linear transformations are maps between vector spaces preserving the two linear operations.
		The notion of linear transformation is difficult to overestimate: it covers such diverse objects as linear systems, (systems of) linear differential/integral equations, derivative, integral, Fourier transform, Laplace transform, etc.
		Linear combinations, linear independence, bases and isomorphisms (concepts, detecting if a vector belongs to a given span and if a given set of vectors is linearly independent, basis, coordinates, isomorphism to R
	www.fen.bilkent.edu.tr /~degt/math220/math220.html (1612 words)

	2.3.1 Linear algebra object creation
		We have the notion of a multiscalar, which is a linear algebra object that exists as a unit entirely within one node (i.e., it is NOT distributed).
		Together with the calling sequence for creating a projected vector, we also give the calling sequence for creating a projected multivector, which is the natural extension of the concept to multivectors:
		However, depending on whether the (multi)vector is projected against a column or row, the roles of the global length and width of the multivector may be reversed.
	www.cs.utexas.edu /~rvdg/pubs/Guide/node34.html (863 words)

	Projection
		In mathematics a projection is a linear transformation which remains unchanged if applied twice (''p''(''u'') = p(''p''(''u''))) (in other words, it is idempotent), such as that taking (''x'', y, z) in three dimensions to (''x'', y, 0) in the plane or generalisations of this in other dimensions.
		In military strategy and international relations, projection of force refers to the latent or actualized capacity to win armed contests in distant theatres.
		To these personal qualifications must be added a great advantage--and a perfect and intimate knowledge of all the recesses various individuals, whether friendly or hostile, with whom he might come circumstances in which he was placed.
	projection.kiwiki.homeip.net (234 words)

	The Math Forum - Math Library - Matrices
		Linear algebra, sometimes disguised as matrix theory, considers sets and functions which preserve linear structure.
		An introduction to cryptography, in particular affine enciphering applied to the alphabet, focusing on the use of linear algebra.
		A project the goal of which is to develop infrastructure and algorithms for the parallel solution of eigenvalue problems.
	mathforum.org /library/topics/matrices (2162 words)

	MAS 2103, Linear Algebra
		The student will be able to apply the concepts of matrices and vectors to solving systems of equations, determining the properties and operations on vectors, and determining the properties and operations on matrices.
		The student will be able to analyze and solve problems by using the algorithms of linear algebra to find the matrix of a transformation with respect to a basis, change a basis, and to use characteristic vectors to find the diagonal matrix which is similar to a given matrix.
		The student will be able to analyze and solve problems by using the algorithms of linear algebra to find the matrix of a transformation with respect to a basis, change a basis, and to use characteristic vectors to find the diagonal matrix which is similar to a given matrix by being able to:
	www.spcollege.edu /clw/math_science/courses/MAS2103.htm (543 words)

	Syllabus for Linear Algebra - Susan Nelson
		This course is designed to introduce the student to the basic notions of linear algebra.
		A linear transformation is on-to-one, onto, or an isomorphism.
		A linear transformation that satisfies given properties and a computation of the action of that transformation on a given vector,
	www.gpc.edu /~snelson/LinearFall02/syllabus.html (865 words)

	Linear Algebra: Activities
		This is a preparation for your efforts on the term project.
		Readers should also be familiar with the concepts of linear independence and the span of a set of vectors.
		A geometric argument is used to find a matrix P that will project vectors from V onto a subspace W of V.
	online.redwoods.cc.ca.us /instruct/darnold/LinAlg/activity.htm (1916 words)

	Egwald Mathematics - Linear Algebra: Vectors
		* cos(ß) is the length of the projection of w = (5, 9) onto v = (12, 2).
		From the diagram above, the vector p obtained by projecting of w = (5, 9) onto v = (12, 2) is p = (6.32, 1.05).
		The projection of w on v is the vector p given by:
	www.egwald.com /linearalgebra/vectors.php (2285 words)

	VII. REVIEW OF LINEAR ALGEBRA*
		The set of m vectors is said to be independent if no one of them can be written as a linear combination of the others.
		A set of m vectors which span a space V of dimension m are said to form a basis for that space.
		In this example, A projects the 3-dimensional vector into the two-dimensional plane.
	courses.temple.edu /economics/notes/algebra/NBUCK1.htm (1359 words)

	projection - OneLook Dictionary Search
		Projection : A Glossary of Mathematical Terms [home, info]
		Phrases that include projection: mercator projection, homolosine projection, equal-area projection, stereographic projection, conformal projection, more...
		Words similar to projection: ejection, expulsion, jut, jutting, project, projectional, protrusion, protuberance, forcing out, ledge, outjet, more...
	www.onelook.com /?w=projection (480 words)

	Liinear Algebra (Site not responding. Last check: 2007-08-12)
		the concept of matrices and their role in linear algebra and applied mathematics.
		Math 206, Linear Algebra I and Math 242, Differential Equations, or Math 241, Calculus IV and Math 244, Linear Analysis I, and a C- or better in Math 248.
		Students should learn the language and methods of linear algebra and some of the most important applications.
	www.calpoly.edu /~math/ugcourses/linearalg.html (283 words)

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