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Topic: Orthogonal basis

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	GeoSci 236: Empirical Orthogonal Functions
		Another important property for a basis is orthogonality (like sines or various frequencies); we would like to account for a certain component of the signal only once.
		The second mode is the one most commonly realized under the constraint of orthogonality to the first one, the third is the most frequently realized pattern that is orthogonal to both higher modes, and so on.
		Hence the term `empirical'; we still have an orthogonal basis, like the Fourier or Legendre bases, but whose members are not chosen based on analytic considerations, but based on maximization of the projection of the data on them.
	geosci.uchicago.edu /~gidon/geosci236/eofs/eofs.html (1596 words)

	Math 343 Class Notes (Site not responding. Last check: 2007-10-26)
		Its orthogonal complement would be all the points on the line that is perpendicular to the plane and that passes through the origin.
		The orthogonal complement of W is a subspace of V. The only vector common to W and its orthogonal complement is the zero vector.
		The orthogonal complement of the orthogonal complement of W is W. dimension(W) + dimension(orthogonal complement of W) = dimension(V).
	www.math.byu.edu /~dsiebert/math343/notes.html (1497 words)

	Orthonormal basis (via CobWeb/3.1 planetlab2.tamu.edu) (Site not responding. Last check: 2007-10-26)
		In mathematics, an orthonormal basis of an inner product space V(i.e., a vector space with an inner product), or in particular of a Hilbert space H, is a set of elements whose span is dense in the space, in which the elements are mutually orthogonal and normal, that is, of magnitude 1.
		An orthonormal basis is not generally a "basis", i.e., it is not generally possible to write every member of the space as a linear combination of finitely many members of an orthonormal basis.
		Note that in the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish the two, the latter bases are also called Hamel bases.
	orthonormal-basis.iqnaut.net.cob-web.org:8888 (563 words)

	Orthogonality and Least Squares (Site not responding. Last check: 2007-10-26)
		The idea is, of course, that orthogonal bases are much easier to work with and lend themselves nicely to performing various computations such as finding vector coordinates with respect to this basis and projecting vectors on various subspaces of a vector space.
		Orthogonal matrices are incredibly important in geometry and physics.
		One of the reasons why, is that orthogonal 2 x 2 matrices with positive determinant represent rigid motions of the plane that keep the origin fixed.
	math.ucsd.edu /~math20f/Fall/Lab8F/Lab8F.html (2299 words)

	No Title (Site not responding. Last check: 2007-10-26)
		If you have a set of mutually orthogonal vectors (an orthogonal set) taken from some vector space, then the set is always a basis (an orthogonal basis) for some subspace of the vector space.
		An orthonormal basis is an orthogonal basis with the additional property that all of the basis vectors are normalized.
		An orthonormal basis for a vector space is very easy to work with, because only dot products are needed to determine the coordinates for any vector in the space, relative to the basis.
	www.sci.wsu.edu /math/faculty/genz/220v/lessons/l11/l11.html (241 words)

	Orthogonal Basis for Range Space of a Matrix (Site not responding. Last check: 2007-10-26)
		Orthogonal Basis for Range Space of a Matrix
		Computes an orthogonal basis for the range space of the integer, real, double-precision or complex matrix x.
		Each of the columns of the matrix y has norm one and the columns are orthogonal; i.e.;
	www.omatrix.com /manual/orth.htm (112 words)

	Orthogonal and Orthonormal (Site not responding. Last check: 2007-10-26)
		Orthogonal is a fancy word for perpendicular, which is a pretty fancy word all by itself.
		A basis is orthonormal, or unitary, if it is orthogonal and all vectors have unit length.
		It's easy to make an orthogonal basis orthonormal; divide each vector by its length; assuming the lengths are contained in your field of scalars.
	www.mathreference.com /la,orth.html (292 words)

	waveform_fit_rpt_1.nb
		The new orthogonal basis functions are on the left, the input to GramSchmidt is on the right.
		There are two points to be learned from Paul's effort: a) using basis functions which are directly related to the data works and b) using GramSchmidt allows flexibility in the choice of basis functions beyond what one might find in a book on mathematical physics.
		Such a basis is shown on the left, with the input functions on the right.
	www.bartol.udel.edu /~seckel/icetop/waveform_fit_rpt_1.html (1433 words)

	Orthonormal basis - Wikipedia, the free encyclopedia
		In mathematics, an orthonormal basis of an inner product space V (i.e., a vector space with an inner product), or in particular of a Hilbert space H, is a set of elements whose span is dense in the space, in which the elements are mutually orthogonal and normal, that is, of magnitude 1.
		An orthogonal basis satisfies the same conditions, without the condition of length 1; it is easy to change the vectors in an orthogonal basis by scalar multiples to get an orthonormal basis, and indeed this is a typical way that an orthonormal basis is constructed, via an orthogonal basis.
		An orthonormal basis of a vector space V makes no sense unless V is given an inner product; a Banach space does not have an orthonormal basis unless it is a Hilbert space.
	en.wikipedia.org /wiki/Orthonormal_basis (617 words)

	Orthogonal basis functions in discrete least squares rational approximation (Site not responding. Last check: 2007-10-26)
		The orthogonality of the basis is important because if the transfer function G
		Even if an orthogonal basis is used, but with respect to the wrong inner product (e.g., the Lebesgue measure on the unit circle) numerical instability can be fatal in practice.
		We show that both approaches lead to an inverse eigenvalue problem, which forms the common framework in which fast and numerically stable algorithms can be designed for the computation of the orthonormal basis.
	www.cs.kuleuven.ac.be /cwis/research/nalag/papers/ade/iccam02A (265 words)

	Orthogonal Function Basis
		Furthermore, assume the basis functions, employed by t, are orthogonal in the image space.
		This is because distinct basis functions are orthogonal.
		Either is a basis for the analytic functions on the interval [-1,1], however, the legendre polynomials form an orthogonal basis.
	www.mathreference.com /la-xf,ofb.html (504 words)

	Review of Chapters 5 and 6 and loose ends
		The dimension of a vector space is the number of vectors in a basis for the space.
		Find a basis for the null space, column space, or row space of a matrix, or subspaces as given in problems 1-10 in Section 4.5, in order to determine dimension.
		The Basis Theorem: Another work-saver, this says that if you already know the dimension of a space, and you have a set with the "right" number of vectors to be a basis (i.e.
	www.math.uiuc.edu /~johnmaki/ch6review.html (1579 words)

	Discrete Math, Seventh Problem Set (July 2) REU 2003
		A ``nice'' basis is one that is ``close'' to being orthogonal in some vague sense.
		Recall the Gram-Schmidt orthogonalization process for obtaining an orthogonal basis for the span of a set of linearly independent vectors.
		Note that the definition is sensitive to order: the same basis vectors in a different order may not form a Lovász-reduced basis.
	people.cs.uchicago.edu /~laci/reu03/notes7 (818 words)

	Orthogonal moments
		This basis set is non-orthogonal and this property is passed onto the Cartesian moments.
		These orthogonal moments have the advantage of needing lower precision to represent differences to the same accuracy as the monomials.
		The orthogonality condition simplifies the reconstruction of the original function from the generated moments.
	homepages.inf.ed.ac.uk /rbf/CVonline/LOCAL_COPIES/SHUTLER3/node9.html (165 words)

	No Title
		The dimension of a vector space is given by the number of vectors in any basis (this definition makes sense because any two bases for a vector space consist of the same number of vectors).
		Note that the rows for the basis are taken from the Echelon matrix and not from the original matrix A (unless A is already in Echelon form).
		The basis of the column space is the subset of S whose vectors are linearly independent.
	math.la.asu.edu /~tracogna/MAT242/review3b (1085 words)

	Walsh Functions
		Clearly the transform mapping the original sequence into the entries of the bottom row is orthogonal (since it was obtained by a succession of orthogonal transformations).
		The discrete Haar wavelet basis is obtained by choosing the second block in each row and the first and second entry on the last row.
		Since all transformations were orthogonal we must have that the collection of vectors corresponding to this choice of patterns is an orthogonal basis of
	www.math.yale.edu /pub/wavelets/software/xwpl/html/manual/node31.html (420 words)

	Linear Algebra (Math 2318) - Vector Spaces - Orthonormal Basis (Site not responding. Last check: 2007-10-26)
		basis. We are going to be looking at a special kind of basis in this section that can arise in an inner product space, and yes it does require an inner product space to construct. However, before we do that we’re going to need to get some preliminary topics out of the way first.
		So, they do form an orthogonal set. To show that they don’t form an orthonormal set we just need to show that at least one of them does not have a norm of 1. For the practice we’ll compute all the norms.
		orthogonal complement of W) and is called the component of u orthogonal to W.
	tutorial.math.lamar.edu /AllBrowsers/2318/OrthonormalBasis.asp (1119 words)

	BASIS_COMPARE - Compare two basis sets.
		We let Q be the M by N matrix whose columns are an orthogonal basis for the span of the vectors U(1:N).
		CVT_BASIS_FLOW extracts a basis from a set of velocity snapshots using CVT methods.
		POD_BASIS_FLOW extracts a basis from a set of velocity snapshots using POD methods.
	www.csit.fsu.edu /~burkardt/f_src/basis_compare/basis_compare.html (511 words)

	Orthogonal Basis and Orthogonalization (Site not responding. Last check: 2007-10-26)
		The single most useful property of these spaces is that they permit the introduction of an orthonormal basis.
		The first and most important way of specifying such a basis is to introduce a Hermitian matrix or operator.
		The second way of specifying such a basis is by means of the Gram-Schmidt orthogonalization process.
	www.math.ohio-state.edu /~gerlach/math/BVtypset/node15.html (156 words)

	r3su05
		Warning: In order to be an orthogonal matrix it is not enough for the columns (respectively rows) to be an orthogonal set; they must satisfy the stronger condition of being orthonormal.
		So, notice that while every orthogonally diagonalizable matrix is diagonalizable, not every diagonalizable matrix is orthogonally diagonalizable, since there are examples of diagonalizable matrices which are not symmetric.
		Note that if the basis has just one element, there is nothing to do in the first of these steps, so you proceed to the second step.
	www.math.okstate.edu /~myersr/3013/review/r3su05/r3su05.html (1501 words)

	The Gram-Schmidt Algorithm - HMC Calculus Tutorial
		It often greatly simplifies calculations to work in an orthogonal basis.
		To obtain an orthonormal basis for an inner product space V, use the Gram-Schmidt algorithm to construct an orthogonal basis.
		Normalizing the vectors in the orthogonal basis, we obtain the orthonormal basis
	www.math.hmc.edu /calculus/tutorials/gramschmidt (244 words)

	The case of an orthogonal basis
		In the special case of an orthogonal basis
		there is no distinction between covariant and contravariant quantities with respect to the expansion coefficients of this basis, and as such we use Latin suffixes to denote them:
		In other words, for the case of an orthogonal basis
	www.tcm.phy.cam.ac.uk /~aam24/PAPERS/paper4/node5.html (84 words)

	orthogonal basis (Site not responding. Last check: 2007-10-26)
		The orthogonal eigenbasis is the "easiest" to work with, but generally to be a basis a set of vectors has to be lin.
		by using these vectors, but they are not orthogonal.
		So, basis vectors don't have to be orthogonal, but they are usually chosen to be.
	www.physicsforums.com /showthread.php?t=72387 (1253 words)

	linf05finprep
		Find a basis for the null space of a linear transformation.
		We find a basis for each eigenspace, and recall that the dimension of each eigenspace is less than or equal to the algebraic multiplicity of that eigenvalue.
		It follows that a basis for this eigenspace is
	www-math.cudenver.edu /~rrosterm/linf05finprep/linf05finprep.html (832 words)

	Approximation via Subspaces: Analysis and Geometry
		basis vectors has been obtained, it is natural to ask: Does the set of such orthonormal elements constitute a basis for the whole space
		There is a sense in which this worry is justified if the vector is a discontinuous function.
		then it is known as Parseval's relation or the completeness relation for the set of orthonormal basis elements.
	www.math.ohio-state.edu /~gerlach/math/BVtypset/node16.html (398 words)

	Orthogonal Basis for a Subspace
		An orthogonal basis can always be made into an orthonormal basis by normalizing each of the vectors.
		In part b of Steps 1-m the orthogonal vector just produced is normalized.
		Continuing through Step m, we conclude that W is an orthonormal basis for S.
	engr.smu.edu /emis/8371/book/chap3/node12.html (351 words)

	Gram.html
		The orthonormal basis we have constructed is closely related to the
		With an orthogonal basis, this is trivial since it is just a sum of inner products times each basis element.
		Just to emphasize the ease of solving the minimization problem once the basis is in hand, here are a couple more examples.
	www.math.sunysb.edu /~scott/mat310.spr05/docs/Grammy (561 words)

	Metrics and minimisation in non-orthogonal basis sets (Site not responding. Last check: 2007-10-26)
		When working with a non-orthogonal basis, care must be taken with notation.
		It is common to use raised and lowered indices to distinguish between vectors and matrices which transform differently; this has been introduced to electronic structure calculations by Ballentine and Kolár[
		Traditionally, the Hamiltonian is taken as covariant and the density matrix as contravariant.
	www.cmmp.ucl.ac.uk /~drb/IJQC/node16.html (214 words)

	The Basis Problem Revisited
		A solution of the basis problem stated above involves what is known as the Gram-Schmidt procedure.
		Now that we know what an orthonormal basis is, we can describe an important class of matrices which we will use heavily later on.
		Example 5 The identity matrix is an orthogonal matrix.
	distance-ed.math.tamu.edu /Math640/chapter3/node15.html (424 words)

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