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Topic: Orthonormal basis

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Hilbert space

Orthogonal

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	PlanetMath: orthonormal (Site not responding. Last check: 2007-11-07)
		A standard application is finding an orthonormal basis for a vector space, such as by Gram-Schmidt orthonormalization.
		Orthonormal bases are computationally simple to work with.
		This is version 7 of orthonormal, born on 2002-01-04, modified 2003-04-19.
	planetmath.org /encyclopedia/Orthonormal.html (119 words)

	Orthonormality - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-07)
		In linear algebra, two vectors v and w are said to be orthonormal if they are both orthogonal (according to a given inner product) and normalized.
		A set of vectors which are pairwise orthonormal is called an orthonormal set.
		A basis which forms an orthonormal set is called an orthonormal basis.
	en.wikipedia.org /wiki/Orthonormal (146 words)

	Orthonormal Bases in Rn
		What makes the standard basis easy to use is that the vectors are mutually orthogonal and are all unit vectors.
		First, if we can find an orthogonal basis, we can always divide each of the basis vectors by their magnitudes to arrive at an orthonormal basis.
		If we want to find an orthonormal basis we just divide each of these vectors by their magnitudes.
	www.ltcconline.net /greenl/courses/203/Vectors/orthonormalBases.htm (316 words)

	Basis (linear algebra) - FreeEncyclopedia (Site not responding. Last check: 2007-11-07)
		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.
	openproxy.ath.cx /ba/Basis_(linear_algebra).html (363 words)

	A Symmetric and Shift-Invariant Wavelet Basis
		This shift-variance can be seen directly from the construction of their bases, equations (2) and (5), because of the change of step sizes among different scales in these definitions.
		Saito and Beylkin [19] suggested using the shell of an orthonormal basis when shift-invariant is required.
		may still be decomposed in the shell of an orthonormal basis much the same way as it was in an orthonormal wavelet basis itself.
	www.cwp.mines.edu /html_reports/lydia95/node5.html (644 words)

	Orthonormal Basis Functions for Modelling Continuous-Time Systems - Akcay, Ninness (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		The contribution of the paper is to establish that the obtained model sets are complete in all of the Hardy spaces H p (); 1 < p < 1 and the right half plane algebra A() provided that a mild condition on the choice of basis poles is satis ed.
		8 the construction of orthonormal basis functions for system i..
		1 A uni ed approach to the synthesis of orthonormal exponentia..
	citeseer.ist.psu.edu /akcay99orthonormal.html (699 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Andy Tjon, Project 2, CAP5725, Computer Graphics I Part A of this project is to compute a right-handed orthonormal basis uvw from a single vector a such that w is parallel to a.
		Part B of this project is to compute a right-handed orthonormal basis uvw from two vectors a and b such that u is parallel to a and w is normal to the planes defined by a and b.
		One of the entered vectors is the first orthonormal basis vector.
	longwood.cs.ucf.edu /~tjie/cap5725/Proj2.txt (335 words)

	Vector Spaces (Site not responding. Last check: 2007-11-07)
		A linear operator applied successively to the members of an orthonormal basis may give a new set of vectors which no longer span the entire space.
		Diagonalizing a matrix means transforming to an orthonormal basis in which the only nonzero matrix elements are on the diagonal, i = j.
		As previously stated, a unitary matrix is an operator that rotates an orthonormal basis into another orthonormal basis.
	landau1.phys.virginia.edu /classes/751.mf1i.fall02/751LinearAlgebra.htm (3071 words)

	Gram-Schmidt (Site not responding. Last check: 2007-11-07)
		For most purposes, the most convenient type of basis for a vector space is orthonormal, i.e., the vectors are unit length and are pairwise orthogonal.
		For the process to succeed in producing an orthonormal set, the given vectors must be linearly independent.
		The result is an orthonormal basis, so the dot product of each pair of vectors is zero and each vector has unit length.
	www.math.neu.edu /~suciu/mth1230/gram-schmidt/g-s.html (158 words)

	Learn more about Dimension in the online encyclopedia. (Site not responding. Last check: 2007-11-07)
		For vector spaces, there is a natural concept of dimension, namely the cardinality of a basis.
		A connected topological manifold is locally homeomorphic to Euclidean n-space, and the number n is called the manifold's dimension.
		Every Hilbert space admits an orthonormal basis, and any two such bases have the same cardinality.
	www.onlineencyclopedia.org /d/di/dimension.html (669 words)

	No Title (Site not responding. Last check: 2007-11-07)
		An orthonormal basis is an orthogonal basis with the additional property that all of the basis vectors are normalized.
		is an orthonormal basis that you should all be very familiar with.
		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)

	[No title] (Site not responding. Last check: 2007-11-07)
		Then any orthonormal subset T of G will meet the sum of E2 and E3 only in E3, and hence the projection of T to E2 will be countable; but the projection of G onto E2 has dense image.
		Let H_2 be the non-separable Hilbert space with orthonormal basis {y_b: b in B} indexed by B. Let K be the linear span of the vectors {x_b + y_b: b in B} in the direct sum H_1 + H_2.
		Then there is no orthonormal set in K whose span is dense in K. In fact: 1) Any orthonormal set in K is countable.
	www.math.niu.edu /~rusin/known-math/00_incoming/innerprod (534 words)

	The Gram-Schmidt Algorithm - HMC Calculus Tutorial
		To obtain an orthonormal basis for an inner product space V, use the Gram-Schmidt algorithm to construct an orthogonal basis.
		For more abstract spaces, however, the existence of an orthonormal basis is not obvious.
		The Gram-Schmidt algorithm is powerful in that it not only guarantees the existence of an orthonormal basis for any inner product space, but actually gives the construction of such a basis.
	www.math.hmc.edu /calculus/tutorials/gramschmidt (244 words)

	A Short Write-up On Wavelets
		(and all its integer offsets) is a Riesz basis of
		The Riesz bases form a multiresolution but they are not (in general) orthonormal and are therefore, not wavelets.
		We wish to find a wavelet basis of functions that are periodic with a period N. The resulting periodic wavelet may be obtained by periodizing the regular wavelet fuction
	www.cimms.ou.edu /~lakshman/Papers/wavelet/wavelet.html (954 words)

	Orthonormal Basis Functions in Time and Frequency Domain: Hambo Transform Theory
		These basis functions are useful for constructing efficient parameterizations and coding of linear systems and signals, as required in, e.g., system identification, system approximation, and adaptive filtering.
		In this paper, the basis functions are derived from a transfer function perspective as well as in a state space setting.
		The generalized basis functions induce signal and system transforms (Hambo transforms), which have proved to be useful analysis tools in various modelling problems.
	epubs.siam.org /sam-bin/dbq/article/40534 (262 words)

	What is a Multiresolution Analysis?
		Wavelet basis representation of certain signals show advantages over the traditional Fourier basis representation both theoretically and practically.
		The MRA concept was initiated by Meyer [12] and Mallat [11], which provides a natural framework for the understanding of wavelet bases.
		Here, I give a brief description of orthonormal, compactly supported wavelet bases; detailed information can be found, for example, in Daubechies [6] and Jawerth and Sweldens [10].
	www.cwp.mines.edu /html_reports/lydia95/node4.html (461 words)

	Wavefunction (Site not responding. Last check: 2007-11-07)
		In fact, there is a far-reaching generalization of this representation, which gives an analogous representation with respect to what we could loosely call a continuously indexed orthonormal basis of a Hilbert space.
		In mathematical terms, such continuous orthonormal bases are referred to as diagonalizations, because mathematically they correspond to representing certain commutative algebras of operators as algebras of multiplication operators.
		The technical details of how this diagonalization is carried out is beyond the scope of this article, but it generalizes the result of linear algebra that a commutative algebra of operators closed under operator adjoint is diagonalized in some orthonormal basis.
	www.sciencedaily.com /encyclopedia/wavefunction (604 words)

	Gamasutra - Features "C++ Data Structures for Rigid-Body Physics" [07.02.99]
		Storing orientation as an orthonormal basis (three mutually orthogonal unit vectors in the form of a 3x3 matrix) uses nine scalars and is less convenient for interpolating orientation; however, it is much more efficient for transforming vectors to and from a local coordinate frame.
		Because of collision detection this seems to be what the orientation is most frequently used for, so I prefer using an orthonormal basis (see Listing 3).
		The conversion from a basis to a quaternion is not terribly expensive and can be done if interpolation is necessary.
	www.gamasutra.com /features/19990702/data_structures_04.htm (316 words)

	[No title]
		Use that to find the projection of v onto the line which is the span of your first basis vector.
		Find an orthonormal basis \+ B for W and then use ideas from class to extend B to an orthonormal ba sis for V. \nCheck that your set is an orthonormal basis for V. Use L inearSolve to find the coordinates of v=
		Let W=Span(\{u_1,..., u_k\}) where the vectors f orm an orthonormal basis for W and let \{u_1,..., u_n\} be the extens ion of this basis to an orthonormal basis for the inner product space \+ V.
	oregonstate.edu /~finchd/342/labs/feb21lab.mws (801 words)

	Connexions - Content - Search Repository (Site not responding. Last check: 2007-11-07)
		The module looks at decomposing signals through orthonormal basis expansion to provide an alternative representation.
		This module discusses the different types of basis that leads up to the definition of an orthonormal basis.
		Examples are given and the useful of the orthonormal basis is discussed.
	cnx.rice.edu /content/search?words=orthonormal (157 words)

	Modelling with Orthonormal Basis Functions (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Abstract: The decomposing description of linear time-invariant infinite-dimensional dynamics in terms of an orthonormal basis is an important part of modern Systems Theory and has a long history in modelling and identification of dynamical systems dating back to the classical work of Lee [19] and Wiener [36].
		This approach is greatest utility when accurate system descriptions are achieved with a small number of basis functions.
		The development of suitable basis functions that reflect the dominant...
	citeseer.ist.psu.edu /akcay99modelling.html (546 words)

	Quantum Mechanics
		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 (3833 words)

	Wavefunction Article, Wavefunction Information (Site not responding. Last check: 2007-11-07)
		Any separable Hilbert space has an orthonormal basis; these bases are not unique however.
		In fact, there is a far-reaching generalization of an orthonormal representation, which gives an analogous representation withrespect to what we could loosely call a continuously indexed orthonormal basis of a Hilbert space.
		The technical details of how this diagonalization is carried out is beyond thescope of this article, but it generalizes the result of linear algebra that a commutative algebra of operators closed underoperator adjoint is diagonalized in some orthonormal basis.
	www.anoca.org /space/representation/wavefunction.html (597 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Given the basis $1, X, X^2, X^3$, run the Gram-Schmidt algorithm to produce an orthonormal basis of $V$.
		We obtain another basis ${\bf v_n}, {\bf v_{n-1}}, \ldots, {\bf v_1}$ by reversing the order of the vectors.
		Suppose that $\alpha$ carries some orthonormal basis to an orthonormal basis.
	www.bath.ac.uk /~masgcs/MA8/t9.txt (380 words)

	Orthonormal basis functions. (Site not responding. Last check: 2007-11-07)
		To this end, orthonormal basis function (OBF's) expansions, such as Laguerre and Kautz bases, have proven useful.
		High-order frequency domain expressions for the model variance are derived as well as upper bounds on the model bias.
		The machinery necessary for analysing the model variance is developed in E-[3] where certain classical Fourier convergence and asymptotic Toeplitz matrix properties are generalized to general orthonormal basis functions.
	www.s3.kth.se /~hjalmars/node8.html (145 words)

	Selection of Best Orthonormal Rational Basis
		This contribution deals with the problem of structure determination for generalized orthonormal basis models used in system identification.
		Since the variance of the estimated model increases with the number of estimated parameters, one objective is to find coordinates, or a basis, for the finite-dimensional subspace giving as compact or parsimonious a system representation as possible.
		In this paper, a best basis algorithm and a coefficient decomposition scheme are derived for the generalized orthonormal rational bases.
	epubs.siam.org /sam-bin/dbq/article/32818 (177 words)

	No Title
		S is linearly independent and forms a basis for Span(S).
		An orthogonal set S of unit vectors is an orthonormal set.
		An orthonormal S is an orthonormal basis for Span(S).
	www.sci.wsu.edu /math/faculty/genz/220v/lessons/l11o/l11o.html (46 words)

	Math 3450 Take Home Exam (Site not responding. Last check: 2007-11-07)
		_____ All vectors in an orthogonal basis have norm of 1.
		S may be an orthonormal basis with one inner product and not even an orthogonal basis with a different inner product.
		Express x in terms of this new orthonormal basis.
	www.apsu.edu /vandergriffj/spring99/3450/test3.html (201 words)

	Course Notes, Isometries (Site not responding. Last check: 2007-11-07)
		In particular, be sure to know the definitions of isometry, odd and even isometries, orthonormal basis, and the last Theorem on 3-D isometries.
		To discuss isometries it is convenient to use orthonormal bases.
		We frequently need to specify one unit vector and extend it to an orthonormal basis.
	www.rpi.edu /~piperb/geometry/notes/set2/set2.html (665 words)

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