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Topic: Rank of a linear operator

	NationMaster - Encyclopedia: Linear operator
		In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it "preserves linear combinations".
		Differentiation is a linear transformation from the space of all differentiable functions to the space of all functions.
		Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations.
	www.nationmaster.com /encyclopedia/Linear-operator (1064 words)

	NationMaster - Encyclopedia: Identity matrix
		In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere.
		In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of nÃ—n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication.
		In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.
	www.nationmaster.com /encyclopedia/Identity-matrix (832 words)

	General Linear Models (GLM)
		The development of the linear regression model in the late 19th century, and the development of correlational methods shortly thereafter, are clearly direct outgrowths of the theory of algebraic invariants.
		Linear combinations of responses reflecting a repeated measure effect (for example, the difference of responses on a measure under differing conditions) can be constructed and tested for significance using either the univariate or multivariate approach to analyzing repeated measures in the general linear model.
		The general implication of the theory of estimability of linear functions is that hypotheses which cannot be expressed as linear combinations of the rows of X (i.e., the combinations of observed levels of the categorical predictor variables) are not estimable, and therefore cannot be tested.
	www.statsoft.com /textbook/stglm.html (13045 words)

	Reference.com/Encyclopedia/Linear map
		The term "linear transformation" is in particularly common use, especially for linear maps from a vector space to itself (endomorphisms).
		In the language of abstract algebra, a linear map is a homomorphism of vector spaces, or a morphism in the category of vector spaces over a given field.
		A linear operator between topological vector spaces, for example normed spaces, may also be continuous and therefore be a continuous linear operator.
	www.reference.com /browse/wiki/Linear_transformation (1665 words)

	GROUPS, THEORY - Online Information article about GROUPS, THEORY
		An operation of a discontinuous group must necessarily be specified analytically by a system of equations of the form x'1=fe(x', x2,..., x„ ; al, a2,..., a.), (s = I, 2,..., n), and the different operations of the group will be given by different sets of values of the parameters a1, a2,...
		Linear A most important class of discontinuous groups are those discon- that arise in this way from the general linear continuous tinuous group in a given set of variables.
		Hence the group may be presented in the form of a group of linear transformations of a single variable generated by the two linear transformations z'=Yizaiz++SRi r' z'=72z+32, a2z272z+32, which correspond to pairs of inversions at AB, BC and BC, CA respectively.
	encyclopedia.jrank.org /GRA_GUI/GROUPS_THEORY.html (8639 words)

	College Mathematiccs Journal
		The difference operator acts as a linear map on the vector space of sequences of real numbers, and it is natural to ask for the periodic points under iteration of this map.
		The points with period n under the difference operator form a subspace, and using a natural conjugacy between the difference operator and the shift map we determine this subspace explicitly, both for the space of ordinary infinite sequences and for bi-infinite sequences.
		The spectral basis for the commutative algebra generated by a linear operator t on a finite dimensional space consists of orthogonal idempotents and related nilpotent elements that can be determined quickly from the minimal polynomial of the operator.
	www.maa.org /pubs/cmj_toc.html (903 words)

	NeuroSolutions: Frequently Asked Questions
		The true power and advantage of neural networks lies in their ability to represent both linear and non-linear relationships and in their ability to learn these relationships directly from the data being modeled.
		This selection operator is based upon the evolutionary principle know as “survival of the fittest”, whereby a chromosome's probability of getting selected is proportional to its fitness.
		This crossover operator picks a random point within the chromosomes, then switches the genes of the two chromosomes at this point to produce two new offspring.
	www.neurosolutions.com /products/ns/faq.html (2789 words)

	Math Markup Language (Chapter 5)
		On account of the linear algebra context, a visually impaired reader may have directed his or her voice synthesis software to render superscripts as vector components.
		For example, the "rank" operator from linear algebra is not included as a pre-defined MathML content element.
		As a rule, content elements other than containers derive part of their semantic meaning from the surrounding context, such as whether an operator is being applied to arguments or not, or whether a bvar element is qualifying an integral, and so on.
	www.w3.org /TR/1998/REC-MathML-19980407/chapter5.html (2031 words)

	14. Glossary
		ufunc / universal function: a ufunc is a callable object which performs operations on all of the elements of its arguments, which can be lists, tuples, or arrays.
		Matrix: The Matrix module defines a subclass Matrix of the UserArray class which is specialized for linear algebra matrices.
		rank: the rank of an array is the number of dimensions it has, or the number of integers in its shape tuple.
	numpy.scipy.org /numpydoc/numpy-15.html (229 words)

	Magnetic field - Wikipedia, the free encyclopedia
		Otherwise, however, this distinction is often ignored, and both quantities are frequently referred to as "the magnetic field." (Some authors call H the auxiliary field, instead.) In linear materials, such as air or free space, the two quantities are linearly related:
		The rotating magnetic field is a key principle in the operation of alternating-current motors.
		A permanent magnet in such a field will rotate so as to maintain its alignment with the external field.
	en.wikipedia.org /wiki/Magnetic_field (2967 words)

	Math Forum Discussions
		and is independent of the representation using rank-one summands.
		TX and XT are of finite rank, and tr(TX)=tr(XT) (routine proof).
		norm) from finite rank operators to all trace class operators.
	www.mathforum.com /kb/message.jspa?messageID=3879058&tstart=0 (375 words)

	Transactions of the American Mathematical Society
		Abstract: We consider a one-dimensional wave equation, which governs the vibrations of a damped string with spatially nonhomogeneous density and damping coefficients.
		These operators are the generators of the dynamical semigroups corresponding to the above boundary-value problems.
		Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Amer.
	www.ams.org /tran/1997-349-11/S0002-9947-97-02044-8/home.html (591 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal
		The term "linear transformation" is in particular common use, especially for linear maps from a vector space to itself (endomorphisms).
		* T is left-invertible, which is to say there exists a linear map S:W → V such that ST is the identity map on V.
		* T is right-invertible, which is to say there exists a linear map S:W → V such that TS is the identity map on V.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=Linear_transformation (1637 words)

	Math Markup Language (Chapter 5)
		In this case, the use of embedded presentation markup allows the author to specify that v should be rendered in boldface.
		In general, this means that embedded content expressions must be semantically meaningful, since rendering of content markup depends on its meaning.
		Thus, in a presentation context, these elements do not have a clearly defined meaning, and hence there is no obvious choice for a rendering.
	www.w3.org /TR/REC-MathML/chapter5.html (2031 words)

	Combining Presentation and Content Markup
		A second reason to use presentation within content markup is that there is a continually growing list of areas of discourse that do not have pre-defined content elements for encoding their objects and operators.
		Here, the semantics of rank have been given using a symbol from an OpenMath [OpenMath2000] content dictionary (CD).
		Using the terminology of Section 4.2.1 Overview of Syntax and Usage, we see that operator, relation, container, constant and symbol elements make sense on their own, while elements of the qualifier and condition type do not.
	www.w3.org /TR/MathML2/chapter5.html (3690 words)

	NULLSPACE Operator
		Next: RANK Operator Up: Operators with Matrix Arguments Previous: Matrix Cofactors
		The basis is provided in a form such that as many upper components as possible are isolated.
		calculates the rank of A. The rank of a matrix expression can also be found more directly by the
	www.uni-koeln.de /REDUCE/reduce/node170.html (155 words)

	Proceedings of the American Mathematical Society
		In the case of a normal bounded linear operator
		M. Berkani, Restriction of an operator to the range of its powers.
		R. Harte, Invertibility and singularity for bounded linear operators; Marcel Dekker.
	www.ams.org /proc/2002-130-06/S0002-9939-01-06291-8/home.html (239 words)

	Math Forum Discussions
		> > > Fundamental Theorem of Linear Algebra is?
		Ahem...I may be wrong but I thought the Fundamental Theorem of Linear
		polynomial of a linear operator is also an annihilating polynomial of
	www.mathforum.org /kb/message.jspa?messageID=265480&tstart=0 (172 words)

	PMD_Solve_1DD
		source : Rank 1 or 2 array of type REAL(kind=R4 or R8).
		In the example below, we assume a 1D problem where each local operator is represented by a Tridiagonal matrix.
		In case of a 2D problem, WestFD and EastFD parameters must be of rank 1 arrays.
	www.idris.fr /data/publications/PMD/docs/PMD_Solve_1DD.html (316 words)

	List of ACoRN Publications (Sorted by Year) (Site not responding. Last check: )
		Ping-pong effects in linear parallel interference cancellation for CDMA.
		A Forward-only Recursion Algorithm for MAP Decoding of Linear Block.
		Optimal Linear Multistage Receivers and the Recursive Large System SIR.
	www.acorn.net.au /db/pubsbyyear.cfm (4655 words)

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