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Topic: Basis of a vector space

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	PlanetMath: basis (Site not responding. Last check: 2007-10-22)
		A (Hamel) basis of a vector space is a linearly independent spanning set.
		This introduces the notion of dimension of a vector space, which is precisely the cardinality of the basis, and is denoted by
		Characterization of basis of finite dimensional vector space
	planetmath.org /encyclopedia/Basis.html (221 words)

	Glossary - Linear Algebra
		The column space of a matrix is the subspace of the codomain which is spanned by the columns of the matrix.
		The row space of a matrix is the subspace of the domain which is spanned by the rows of the matrix.
		A scalar is an element of the field of scalars for a vector space.
	www.math.umbc.edu /~campbell/Math221/Glossary (1250 words)

	Basis (linear algebra) - Wikipedia, the free encyclopedia
		All bases of a vector space have the same cardinality (number of elements), called the dimension of the vector space.
		An orthonormal basis of an infinite-dimensional Hilbert space is therefore not a Hamel basis.
		In topological vector spaces, quite generally, one may define infinite sums (infinite series) and express elements of the space as certain infinite linear combinations of other elements.
	en.wikipedia.org /wiki/Basis_(linear_algebra) (1033 words)

	Talk:Dual space - Wikipedia, the free encyclopedia
		General tensors on a vector space V are built up from the tensor product of some copies of V and its dual space.
		The article claims that the continuous dual of a normed vector space is something stronger: a Banach space.
		It's probably assuming that the base field of a normed vector space must be R or C.
	en.wikipedia.org /wiki/Talk:Dual_space (326 words)

	Functional analysis - Wikipedia, the free encyclopedia
		In the modern view, functional analysis is seen as the study of complete normed vector spaces over the real or complex numbers.
		These spaces are of fundamental importance in the mathematical formulation of quantum mechanics.
		In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear functionals.
	en.wikipedia.org /wiki/Functional_analysis (708 words)

	No Title (Site not responding. Last check: 2007-10-22)
		A set of basis vectors must be independent, because if they were not, one of them could be written as a linear combination of the others; such a vector could then be removed from the original set to form a smaller spanning set for the subspace.
		Once a basis is available for a subspace, every vector in that subspace has a unique representation as a linear combination of the basis vectors.
		The coordinates of some vector, relative to a basis, are the multipliers that would be used to express the original vector as a linear combination of the basis vectors.
	www.sci.wsu.edu /math/faculty/genz/220v/lessons/l08/l08.html (381 words)

	Encyclopedia: Functional analysis (Site not responding. Last check: 2007-10-22)
		In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces.
		In functional analysis, a normal operator on a Hilbert space H is a continuous linear operator N : H → H that commutes with its hermitian adjoint N: N N = N* N. The main importance of this concept is that the spectral theorem applies to normal operators.
		In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...
	www.nationmaster.com /encyclopedia/Functional-analysis (2211 words)

	PlanetMath: every vector space has a basis (Site not responding. Last check: 2007-10-22)
		This result, trivial in the finite case, is in fact rather surprising when one thinks of infinite dimensionial vector spaces, and the definition of a basis: just try to imagine a basis of the vector space of all continuous mappings
		Cross-references: span, maximal element, vectors, collection, upper bound, chain, inclusion, subset, linearly independent, field, Zorn's lemma, axioms, axiom of choice, equivalent, theorem, mappings, continuous, basis, vector spaces, infinite, finite
		This is version 5 of every vector space has a basis, born on 2002-09-30, modified 2005-06-24.
	planetmath.org /encyclopedia/EveryVectorSpaceHasABasis.html (184 words)

	Moving frame article - Moving frame mathematics smooth manifolds vector space basis tangent space - What-Means.com (Site not responding. Last check: 2007-10-22)
		Given such a manifold M and a point P in it, a frame at P means a vector space basis of the tangent space to M at P.
		In contrast in special relativity M is taken to be a vector space V (of dimension four).
		For example, given a curve in space, the first three derivative vectors of the curve can in general give a frame at a point of it (cf.
	www.what-means.com /encyclopedia/Frames (485 words)

	Null space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-22)
		The null space (also nullspace) of a (A rectangular array of elements (or entries) set out by rows and columns) matrix A is the set of all vectors v which solve the equation Av = 0.
		It is a (Click link for more info and facts about linear subspace) linear subspace of the space of all vectors.
		That means that the null space basis vector is [-1 1].
	www.absoluteastronomy.com /encyclopedia/n/nu/null_space.htm (559 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-22)
		For the first question, part (a), the vectors are not bases if you can express one of them in terms of the other two, i.e.
		If they are consistent one vector is a linear combination of the other two and so the vectors could not form a basis for R^3.
		Finally, for the third problem, any vector not in the plane of v1 and v2 could form the third vector of the basis.
	mathforum.org /library/drmath/view/51987.html (541 words)

	No Title (Site not responding. Last check: 2007-10-22)
		A vector is normalized by dividing by the vector by its length, so that the result is a unit vector.
		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)

	Algebra:Vector spaces - Wikibooks
		A vector space is a way of generalizing the concept of a set of vectors.
		The vector space is a "space" of such abstract objects, which we term "vectors".
		When we look at various vector spaces, it is often useful to examine their subspaces.
	en.wikibooks.org /wiki/Algebra:Vector_spaces (814 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		In that context, the "usual" basis is called a Hamel basis: it is a collection of vectors {v_i}, i in some index set, such that every vector in the space can be written uniquely as a linear combination of some finite set of v_i's.
		For a Schauder basis, on the other hand, you allow an infinite "linear combination", where the infinite sum is understood as an infinite series, converging in the norm topology of the Banach space.
		For the space of all continuous functions on [0,1], normed by the supremum norm, there is an explicit basis: the binary roof functions.
	www.math.niu.edu /~rusin/papers/known-math/96/schauder (766 words)

	Linear Algebra, Fall 1999 (Site not responding. Last check: 2007-10-22)
		Be able to fill in the explanations in the proof that any two basis of a vector space contain the same number of vectors (see handout).
		Be able to explain why any linearly independent set of vectors in a vector space can be extended to a basis for the vector space.
		Be able to find the dimension of the null space and the dimension of the column space for a given matrix.
	www.emu.edu /courses/math351/obj9.htm (139 words)

	Basis (linear algebra) - FreeEncyclopedia (Site not responding. Last check: 2007-10-22)
		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)

	The SUIFMATH Library - Vector Space Utilities (Site not responding. Last check: 2007-10-22)
		method increases the space dimension of the vector space to the value given as its argument.
		The argument is factored with the LU decomposition of the vector space's matrix representation.
		If the resulting factored vector is dependent, then it is already in the vector space.
	suif.stanford.edu /suif/suif1/docs/suifmath_30.html (159 words)

	math405 (Site not responding. Last check: 2007-10-22)
		Abstract vector spaces, bases and dimension (Chapter 2).
		Axioms and properties of an abstract vector space.
		Especially I want everyone to know how the matrix of a linear transformation changes when the basis of the vector space is changed.
	www.math.umd.edu /~jjm/math405.html (298 words)

	Definitions
		Vector space You do not need to memorize this definition, but you must understand it.
		Basis for a vector space or a subspace.
		Orthogonal (or orthonormal) basis for a vector space or subspace.
	www.math.mtu.edu /~msgocken/ma2330spring2001/study3/node2.html (158 words)

	Course Syllabus - Lee College (Site not responding. Last check: 2007-10-22)
		To determine a basis for a vector space.
		To determine the dimension of a vector space.
		To determine an orthonormal basis for a vector space.
	www.lee.edu /syllabus_descr.asp?CRSE_ID=001836 (796 words)

	Untitled (Site not responding. Last check: 2007-10-22)
		In a limited context, the terms "Covariance" and "Contravariance" arise in cases where you have a basis for a vector space that is NOT orthonormal.
		In the restricted context of the presence of a metric, one way to view covariance and contravariance is to think of them as different basis descriptions of the same object.
		In actuality though, vectors and covectors are more generally defined than in the above conceptual model, but in the case of the existence of a metric, the model is equivalent and will take you a long way.
	home.pacbell.net /bbowen/covariant.htm (610 words)

	GAP Manual: 40 Matrix Algebras
		For example the computation of a vector space basis, of coefficients with respect to such a basis, and of representatives under the action on a vector space by right multiplication.
		And the concept of vector space bases is the same as that for row spaces (see Bases for Matrix Algebras).
		The default way to compute a vector space basis of a matrix algebra from a set of generating matrices is to compute this standard basis and a semi-echelonized basis in parallel.
	www.maths.may.ie /staff/jmurray/gap_manual/CHAP040.htm (1071 words)

	Math 225, Review for Test #3 (Site not responding. Last check: 2007-10-22)
		prove that a given subset of a vector space is or is not a subspace
		Theorems 4.1, 4.2, 4.3 (span, null space and column space are subspaces)
		be able to prove that a subset H of a vector space V is a subspace
	www.math.uiuc.edu /~dcmurphy/math225/review3.html (373 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-22)
		A basis is a linearly independent set of vectors that "span" a space.
		In other words, for a set of vectors to form a basis, you have to be able to express any ordered triple (x,y,z) in R^3 as a linear combination of those vectors, and you can't be able to express a vector in the set as a combination of other vectors in the set.
		One of the vectors is a linear combination of the others, so this is not a basis.
	mathforum.org /library/drmath/view/51967.html (534 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Show a given set of vectors is a basis for a given vector space (or not 9.
		Find the dimension of the vector space spanned a set of vectors b) Find a basis for the row space of A and hence the dimension of the row space.
		b) For a given linear transformation determine its: Matrix representation A, kernel (null space of matrix A), basis and dimension of the kernel, and a basis for its range (basis for column space of the matrix).
	www.saintjoe.edu /~karend/m244/m244rf-032.doc (524 words)

	Bases of Vector Spaces, the Basis Problem
		We now know that finitely generated vector spaces which are generated by a set of nonzero vectors have bases and are finite-dimensional.
		Clearly, finite-dimensional vector spaces are finitely generated by a basis.
		After all, each element of the code is a sum of 30 basis vectors, and there are 2 choices for each of the 30 coefficients.
	distance-ed.math.tamu.edu /Math640/chapter3/node11.html (1642 words)

	Gram-Schmidt (Site not responding. Last check: 2007-10-22)
		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.
		If the given vectors are not linearly independent, indeterminate or zero vectors may be produced.
		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)

	[No title] (Site not responding. Last check: 2007-10-22)
		To check whether a subset W of a vector space V is a subspace, it suffices to show the zero vector is in W, and W is closed under addition and scalar multiplication.
		If V is a finite-dimensional vector space, then every basis for V has the same number of elements.
		Suppose V is a finite-dimensional vector space with subspace W.
	www.sju.edu /~amyers/TEACHING/MATH312/312Midterm.html (251 words)

	Basis (Site not responding. Last check: 2007-10-22)
		Given a vector space, a basis is a maximal set of linearly independent vectors.
		Suppose s is a set of vectors that forms a basis, and y is a vector that is not spanned by s.
		A coefficient from r is applied to the first vector, and independent of that, a coefficient is applied to the second vector, and a coefficient is applied to the third vector, and so on.
	www.mathreference.com /la,basis.html (500 words)

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