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Topic: Linear basis

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Basis transformation matrix

	PlanetMath: linear algebra (Site not responding. Last check: 2007-10-12)
		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 (473 words)

	Kids.Net.Au - Encyclopedia > Linear algebra Basis for a vector space (Site not responding. Last check: 2007-10-12)
		A basis of a vector space is sometimes called a Hamel basis in order to distinguish it from the concept of an orthonormal basis of a Hilbert space and some other kinds of bases that occur in Banach spaces.
		An orthonormal basis of a Hilbert space H is an orthonormal set of members of H such that any member of the H can be written as a linear combination of a possibly infinite set of members of the orthonormal basis.
		Every basis of a vector space has the same cardinality, called the dimension of the vector space.
	www.kids.net.au /encyclopedia-wiki/li/Linear_algebra___Basis_for_a_vector_space (416 words)

	PlanetMath: semilinear transformation
		Every linear transformation is semilinear, but the converse is generally not true.
		By the Freshman's dream it is known that this is a field automorphism.
		Thus semilinear maps are useful because they define the automorphism group of the projective geometry of a vector space.
	planetmath.org /encyclopedia/SemilinearTransformation.html (287 words)

	Linear Transformations
		If you want postmultiplication to be a linear map, our vector space is more than a left r module, it is a bimodule, with r on either side.
		Just to confuse you, a linear transformation is also called a linear function, or a linear map, or a linear operator, or a linear transform.
		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.
	www.mathreference.com /la,xform.html (959 words)

	Differential Forms and the Generalized Stokes Theorem
		A remarkable fact is that any vector space has a basis, meaning that there exists a subset of vectors in the space such that any vector in the space can be represented as a linear combination of members of the basis.
		What these definitions of addition of linear functionals and scalar multiplication of linear functions is that the set of linear functionals, with the scalar field and the definitions of addition and multiplication constitute a vector space.
		The vector space of linear functionals over V is said to be dual to the vector space involving V. For vector spaces with finite bases the dual spaces are not very exotic; they are essentially the same as the original spaces.
	www.sjsu.edu /faculty/watkins/difforms.htm (1932 words)

	Crash option
		Except on restarts, a CRASH procedure is used to select an initial basis from certain rows and columns of the constraint matrix (A - I).
		The first call looks for a triangular basis in linear rows, and the iteration proceeds with simplex iterations until the linear constraints are satisfied.
		The Jacobian is then evaluated for the first major iteration and CRASH is called again to find a triangular basis in the nonlinear rows (retaining the current basis for linear rows).
	www.stanford.edu /group/SOL/software/snoptHelp/The_SPECS_file/Description_of_optional_parameters/snopt_Specs_Description_Crash_option.htm (420 words)

	Principles of Virus Architecture
		In this way it is possible to build a variety of complicated bodies in which 5:3:2 symmetry is preserved and in which the number of subunits is a multiple of 60 as predicted by Crick and Watson.
		The theoretical basis for the structure of isometric viruses was put on a firm foundation by Caspar and Klug (1962) with their concept of identical elements in quasi-equivalent environments.
		"Linear" viral capsids have RNA genomes that are encased in a helix of identical protein subunits.
	www.uct.ac.za /depts/mmi/stannard/virarch.html (1262 words)

	Math through DSP: Basic linear algebra
		Given two vector spaces, a basis in the first and a set of vectors of equal cardinality in the second, we can always construct a linear mapping which maps each of the basis vectors to a corresponding vector in the other set.
		Given two vector spaces and a basis for the first, a homomorphism is completely characterized by the set of images of the basis vectors.
		Notice that in a change of basis, the linear mappings stay the same but their representation with respect to the two bases (that is, transform coefficients) change.
	www.helsinki.fi /~ssyreeni/dsound/dspmath-c-04 (2838 words)

	General Conditions
		is also a linear combination of the original vectors, and hence is in the vector space.
		Theorem: The linear combination expressing a vector in terms of basis vectors for a vector space is unique.
		Note that while the linear combination relative to a particular basis is unique, the choice of basis vectors is not.
	www.technick.net /public/code/cp_dpage.php?aiocp_dp=guide_dft_general_conditions (822 words)

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

	Grade Eight - Content Standards (CA Dept of Education)
		As a basis for understanding this concept and addressing the content in the other three strands, students should develop their own questions and perform investigations.
		Recognize the slope of the linear graph as the constant in the relationship y=kx and apply this principle in interpreting graphs constructed from data.
		Distinguish between linear and nonlinear relationships on a graph of data.
	www.cde.ca.gov /be/st/ss/scgrade8.asp (1102 words)

	Wikinfo \| Basis (linear algebra)
		In mathematics, a subset S of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions:
		This definition includes a finiteness condition: a linear combination is always a finite sum of the form a
		Images, some of which are used under the doctrine of Fair use or used with permission, may not be available.
	www.wikinfo.org /wiki.php?title=Basis_(linear_algebra) (445 words)

	math lessons - Basis (linear algebra)
		In mathematics, a basis is a minimum set of vectors that, when combined, can address every vector in a given space.
		The phrase Hamel basis is sometimes used to refer to a basis as defined above, in which the fact that all linear combinations are finite is crucial.
		What is called an orthonormal basis is a set of mutually orthogonal unit vectors that "span" the space via sometimes-infinite linear combinations.
	www.mathdaily.com /lessons/Basis_(linear_algebra) (838 words)

	Changing basis in order to diagonalize a matrix
		In this problem, you are asked to find a new basis in which the matrix A is diagonalized.
		Suppose that the matrix A is a linear operator on R(2) represented in the standard basis {i,j}, by the matrix
		linear algebra eigen values - Find eigenvalues and eigenvectors of the matrix A=(2 1 9 2) By transforming the matrix in the basis of eigenvectors, show explictly that the matrix can diagonalized in the eigen vector basis.
	www.brainmass.com /homeworkhelp/math/algebra/10040 (312 words)

	Linear Algebra - Preliminary lectures
		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.
		The new basis formed by eigenvectors is the best basis one can choose for expressing the corresponding linear transform in a simple expression.
		Since the matrix corresponding to the linear transform is diagonal and since each eigenvector corresponds to a dimension (i.e.
	vision.unige.ch /~marchand/teaching/linalg (1819 words)

	EconPapers: Basis- and tripartition identification for quadratic programming and linear complementarity problems: from ...
		A basis identification algorithm is an algorithm which generates a complementary basis, starting from any complementary solution.
		In linear programming such algorithms were respectively proposed by Megiddo in 1991 and Balinski and Tucker in 1969.
		The presented algorithms are based on the principal pivot transform and the orthogonality property of basis tableaus.
	econpapers.repec.org /paper/dgreureir/199727.htm (270 words)

	Egwald Mathematics - Linear Algebra: Vectors
		Use the Gram-Schmidt process to obtain an orthogonal basis for the two vectors that span the im(T) subspace.
		It is desirable to obtain an orthonormal basis for a vector subspace because it is easy to work with an orthonormal basis.
		On the matrices web page, the Gram-Schmidt process is used to construct an orthonormal basis from a set of linearly independent vectors.
	www.egwald.com /linearalgebra/vectors.php (2285 words)

	1 and i form a linear basis for R(i) (Site not responding. Last check: 2007-10-12)
		Prove that 1, i = sqrt{-1} form a linear basis for R(i), where R is the real field.
		Show that g = sqrt{2} - 2i is algebraic relative to R by finding a polynomial having g as a zero.
		Re: 1 and i form a linear basis for R(i)
	www.talkabouteducation.com /group/alt.algebra.help/messages/62762.html (342 words)

	Re: Linear Algebra concept question
		The two kinds of projection you mention are different if the basis is not orthonormal.
		If you use the basis {(1, 0), (1, 1)}, then the projection in the second sense is the zero vector.
		On R^n, with respect to the standard basis, an inner product can be represented by a positive-definite symmetric n-by-n matrix, and there are clearly many such matrices.
	www.usenet.com /newsgroups/sci.math/msg22632.html (470 words)

	AMS Glossary
		—An approximation to a differential equation that uses a linear combination of basis functions with weights determined by minimizing a globally integrated measure of the error in the solution.
		The resulting error is orthogonal to each basis function in an integral sense.
		Spectral approximations are produced when spherical harmonic basis functions are selected over spherical domains.
	amsglossary.allenpress.com /glossary/browse?s=G&p=1 (329 words)

	FEM1D_NONLINEAR - Finite Element Method for 1D Nonlinear Problem (Site not responding. Last check: 2007-10-12)
		FEM_BASIS_T3_DISPLAY is a MATLAB program which displays a basis function associated with a linear triangle ("T3") mesh.
		FEM1D_ADAPTIVE is a C++ program that applies the finite element method to a linear two point boundary value problem in a 1D region, using adaptive refinement to improve the solution.
		NSASM is a C library of routines, intended to be used with a MATLAB calling program, and which set up the sparse matrix needed for a Newton iteration to solve a finite element formulation of the steady incompressible 2D Navier Stokes equations.
	www.csit.fsu.edu /~burkardt/cpp_src/fem1d_nonlinear/fem1d_nonlinear.html (902 words)

	SONICS: Statistical Analysis Primer
		Per-Minute Statistics - Another important breakthrough for analysis of the NBA was finding that statistics calculated on a per-minute basis tend to be fairly consistent even when a player changes his role and begins to play more minutes.
		Linear weights - The most common way of evaluating players' overall ability is through the use of what's known as "linear weights" formulas, so named because they assign a weight to each statistic (rebounds, steals, points, etc.) and add or subtract them.
		The most commonly used linear weights are the NBA.com Efficiency System and John Hollinger's Player Efficiency Rating (PER).
	www.nba.com /sonics/news/stats101.html (1366 words)

	2.4 Selection of the Basis (Site not responding. Last check: 2007-10-12)
		The resultant basis set, which is said to be kineticly balanced, will be able to fulfill the low field boundary conditions without requiring mixture of the positive and negative energy solutions.
		Once a kineticly balanced basis set has been constructed and the scalar integrals have been evaluated, the DHF equations may be implemented in a straightforward manner.
		In order to take advantage of this type of symmetry, symmetry adapted basis functions which transform as a particular irreducible representation of the double group of interest are constructed from linear combinations of the original basis functions.
	zopyros.ccqc.uga.edu /~kellogg/docs/rltvt/node11.html (708 words)

	Linear Algebra Syllabus (Site not responding. Last check: 2007-10-12)
		The essential topics of linear algebra are prerequisite for many of the subsequent mathematics courses.
		In particular, certain linear algebra concepts are decidedly useful in multivariable calculus, differential equations and statistics.
		The purpose of the course is to provide an introduction to linear algebra at an elementary level to improve students' ability of abstract reasoning by its attention to mathematical proof.
	www.rowan.edu /open/depts/math/Syllbabi/UND.COURSES/LinAlg.html (220 words)

	Quantum Mechanics (Stanford Encyclopedia of Philosophy)
		In classical theories, a set of quantities which forms a supervenience basis for the rest is typically designated as ‘basic’ or ‘fundamental’, and, since any mathematically possible way of combining their values is a physical possibility, the state-space can be obtained by simply taking these as coordinates.
		So, for instance, the state-space of a classical mechanical system composed of n particles, obtained by specifying the values of 6n real-valued quantities — three components of position, and three of momentum for each particle in the system — is a 6n-dimensional coordinate space.
		Different operators can have different eigenvectors, but the eigenvector/operator relation depends only on the operator and vectors in question, and not on the particular basis in which they are expressed; the eigenvector/operator relation is, that is to say, invariant under change of basis.
	plato.stanford.edu /entries/qm (3823 words)

	Basis (Site not responding. Last check: 2007-10-12)
		Given a vector space, a basis is a maximal set of linearly independent vectors.
		If a linear combination of vectors drawn from any of these sets is 0, those vectors are all present in one of the sets in c.
		Suppose s is a set of vectors that forms a basis, and y is a vector that is not spanned by s.
	www.mathreference.com /la,basis.html (500 words)

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