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	Vector (spatial) - Wikipedia, the free encyclopedia
		A common example of a vector is force — it has a magnitude and an orientation in three dimensions (or however many spatial dimensions one has), and multiple forces sum according to the parallelogram law.
		Examples are "moving north at 90 km/h" or "pulling towards the center of Earth with a force of 70 newtons".
		Also, let, for example, a vector field be expressed as three space coordinate functions of three variables, and apply the formula for the curl based on these functions, resulting in three additional functions, which represent a second vector field.
	en.wikipedia.org /wiki/Vector_(spatial) (2834 words)

	Vector space - Open Encyclopedia (Site not responding. Last check: 2007-11-04)
		The fundamental concept in linear algebra is that of a vector space or linear space.
		Given a vector space V, any nonempty subset W of V which is closed under addition and scalar multiplication is called a subspace of V. It is easy to see that subspaces of V are vector spaces (over the same field) in their own right.
		Given two vector spaces V and W over the same field F, one can define linear transformations or “linear maps” from V to W. These are maps from V to W which are compatible with the relevant structure—i.e., they preserve sums and scalar products.
	open-encyclopedia.com /Vector_space (1073 words)

	Examples of vector spaces -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-04)
		The space of all (A mathematical relation such that each element of one set is associated with at least one element of another set) functions from X to V is a vector space over F with coordinate-wise addition and multiplication.
		An important example arising in the context of (The part of algebra that deals with the theory of linear equations and linear transformation) linear algebra itself is the vector space of (Click link for more info and facts about linear transformation) linear transformations.
		For example the (A number of the form a+bi where a and b are real numbers and i is the square root of -1) complex numbers C form a two dimensional vector space over the real numbers R.
	www.absoluteastronomy.com /encyclopedia/e/ex/examples_of_vector_spaces.htm (1560 words)

	Linear Vector Spaces (Site not responding. Last check: 2007-11-04)
		The space of ordinary vectors in three-dimensional space is 3-dimensional.
		is a subspace of the space of ordinary vectors in 3 dimensions.
		Vector addition is different from ordinary addition, but it obeys the rules for the addition operation of a vector space.
	electron6.phys.utk.edu /qm1/modules/m3/Vector_space.htm (1211 words)

	Learn more about Vector (spatial) in the online encyclopedia. (Site not responding. Last check: 2007-11-04)
		Such a vector is a special case of a tensor and is also analogous to a four-vector in relativity (and is sometimes therefore called a three-vector in reference to the three spatial dimensions, although this term also has another meaning for p-vectors of differential geometry).
		Informally, a vector is a quantity, characterized by a number (indicating size or "magnitude") and a direction, that is often represented graphically by an arrow.
		Examples are "moving north at 90 m.p.h" or "pulling towards the center of Earth with a force of 70 Newtons".
	www.onlineencyclopedia.org /v/ve/vector__spatial_.html (1961 words)

	Read about Vector space at WorldVillage Encyclopedia. Research Vector space and learn about Vector space here! (Site not responding. Last check: 2007-11-04)
		A vector space (or linear space) is the basic object of study in the branch of
		vectors, and the operations one can perform upon these vectors such as addition of vectors, scalar multiplication, with some natural constraints such as closure of these operations, associativity of these and combinations of these operations, and so on, we arrive at a description of a mathematical structure which we call a vector space.
		To determine if a set V is a vector space, one only has to specify the set V, a field F, and define vector addition and scalar multiplication in V.
	encyclopedia.worldvillage.com /s/b/Vector_space (975 words)

	Vector Spaces
		It is important that a real vector space consist of the set of vectors and the two operations with certain properties.
		To show that an object is a vector space, we must show that closure for both operations holds [parts (b) and (c) of the definition] and that properties 1 through 8 hold.
		When you are working with complex vector spaces, it is important to remember that the vectors can be constructed by using complex numbers and that the scalars for scalar multiplication can be any complex number.
	distance-ed.math.tamu.edu /Math640/chapter3/node4.html (1048 words)

	Ring theory
		Important examples of commutative rings can be constructed as rings of polynomials and their factor rings.
		Non-commutative rings and associative algebras (rings that are also vector spaces) are often studied via their categories of modules.
		Examples of non-commutative rings are given by rings of square matrices or more generally by rings of endomorphisms of abelian groups or modules, and by monoid rings.
	www.brainyencyclopedia.com /encyclopedia/r/ri/ring_theory.html (674 words)

	Class Notes for Exam 3 (Site not responding. Last check: 2007-11-04)
		W is a vector space under the same vector addition and scalar multiplication used in V (i.e., W meets the 10 conditions necessary for being a vector space outlined in Section 5.1 of the text).
		Row Space: The row space of A is the subspace of n-space spanned by the rows in A. Column Space: The column space of A is the subspace of m-space spanned by the columns of A. Example: Consider the matrix
		A second example of an orthogonal complement is the relationship between the row space and null space of a m x n matrix A. In particular, these two subspaces of n-space are orthgonal complements.
	www.math.byu.edu /~dsiebert/math343/prevnotes3.html (3326 words)

	ALGEBRA - contents (Site not responding. Last check: 2007-11-04)
		Examples of isomorphic and of non-isomorphic pairs of fields.
		Vector spaces: definition of a vector space over a field.
		as the dimension of the space spanned by rows.
	alpha.mini.pw.edu.pl /~tomtracz/alg/topics.html (259 words)

	OBJ2 (Site not responding. Last check: 2007-11-04)
		Be able to verify whether a given set of "objects" form a vector space, i.e., understand the definition of a vector space.
		Be able to determine whether a given subset of a vector space V is a subspace of V.
		Be able to define the null space of a matrix A and the column space of a matrix A. Given a matrix A and a vector v, be able to determine if v is in the null space of A, or if v is in the column space of A, or neither.
	www.emu.edu /courses/math351/obj7.htm (277 words)

	[ref] 59 Vector Spaces
		Examples are non-Gaussian row and matrix spaces, and subspaces of finite fields and abelian number fields that are themselves not fields.
		Storing the module allows one for example to deal with bases whose basis vectors have not yet been computed yet (see Basis); furthermore, in some cases it is convenient to test membership of a vector in the module before computing coefficients w.r.t.
		Examples of such vector spaces are vector spaces of field elements (but not the fields themselves) and non-Gaussian row and matrix spaces (see IsGaussianSpace).
	www.gap-system.org /Manuals/doc/htm/ref/CHAP059.htm (4482 words)

	MC147 Introductory Linear Algebra
		The concept of a field, introduced in MC145, is used in defining an abstract vector space.
		This course begins by defining vector spaces over a field and introduces the ideas which lead to the concept of dimension.
		Definition of a vector space over a field F, fundamental examples of vector spaces, elementary consequences of the definition, subspaces, intersections of subspaces, linear combinations of vectors, the space spanned by a set of vectors, linear independence, basis, dimension, the exchange process
	www.mcs.le.ac.uk /Modules/Year2/MC147.html (664 words)

	Linear Vector Spaces
		The norm of a vector is related to the notion of the length of a vector.
		An inner product space is a linear vector space in which an inner product can be defined for all elements of the space and a norm is given by equation 3.
		The distance between two vectors is taken to be the norm of the difference of the vectors.
	cnx.rice.edu /content/m11236/latest (1115 words)

	TRANS Nr. 15: Bernhard Lauth (University of Munich, Germany): Transtheoretical structures in the natural and social ...
		Vector spaces play a crucial role in almost all areas of classical and non-classical physics, from Newtonian mechanics to Einstein's special and general relativity.
		The physical state of a mechanical system with n degrees of freedom, say, is completely determined by its canonical (position and momentum) coordinates, hence by the position of the system in a 2n-dimensional vector space (the phase space of the system, which should not be confused with the ordinary three-dimensional space).
		For example, recursively enumerable languages correspond to Turing machines, regular languages to finite automata and context-free languages to pushdown automata respectively(5).
	www.inst.at /trans/15Nr/01_6/lauth15.htm (2763 words)

	[No title]
		Then the vector $c_1v_1+\cdots +c_nv_n$ is called a {\em linear combination} of $v_1,\ldots,v_n$.\\ \noindent {\bf 1.5 Definition (Subspace)} A subset of a vector space $V$ which is itself a vector space with respect to the operations in $V$ is called a {\em subspace} of $V$.
		The interchange of rows 2 and 5, for example, is recorded by interchanging the integers 2 and 5 in this vector.
		Thus if one finds the permutation vector $S$ (which is returned when the LU factorization is completed) then the components of the vector $c=Pb$ can be retrieved from $b$ via the vector $S$, which is used as a `pointer'.
	www.math.uab.edu /chernov/teaching/631notes (7043 words)

	Linear Algebra - Preliminary lectures (Site not responding. Last check: 2007-11-04)
		A fundamental notion is that of vectors which in turn define vector spaces.
		The restriction of a vector space leads to a vector subspace.
		This definition goes down to saying that multiplying two vectors from a given subspace by any two scalars and adding the newly formed vectors together results in a vector which lies in the initial subspace.
	vision.unige.ch /~marchand/teaching/linalg (1819 words)

	Abstract linear spaces
		The parallel development in analysis was to move from spaces of concrete objects such as sequence spaces towards abstract linear spaces.
		Bellavitis then defines the 'equipollent sum of line segments' and obtains an 'equipollent calculus' which is essentially a vector space.
		Hamilton represented the complex numbers as a two dimensional vector space over the reals although of course he did not use these general abstract terms.
	www-groups.dcs.st-andrews.ac.uk /~history/HistTopics/Abstract_linear_spaces.html (1865 words)

	[No title] (Site not responding. Last check: 2007-11-04)
		Definitions and examples of vector spaces: Examples such as vectors in space, matrices, functions, polynomials.
		Examples for linear transformations such as rotations, reflections, differentiation, left multiplication.
		Is the derivative operator a linear transformation on V? Define the vector space of 3rd order polynomials and write the matrix of the derivative operator with respect to the basis { 1, (x+1), x^2-x+1, x^3-x^2+x-1 } What is the matrix of the transformation P(x) (P’’(x)+2 P’(x) — P(x).
	www.gursey.gov.tr /week1.doc (420 words)

	Linear Algebra -- Exam 1 Review (Site not responding. Last check: 2007-11-04)
		The axioms for vector spaces -- showing that a set is or is not a vector space using the axioms
		Subspaces of a vector space -- know how to show that a subset of a vector space is or is not a subspace using the ``shortcut test'' (1.2.8)
		The span of any set S in a vector space V is a vector subspace of V.
	math.holycross.edu /~little/LA04/LARev1.html (322 words)

	MA1152 Introductory Linear Algebra
		To understand how to use elementary row operations on matrices to solve systems of linear equations; to understand the role of the rank and the row-reduced echelon form; to understand the varying nature of solution sets.
		The concept of a field, introduced in MA1102, is used in defining an abstract vector space.
		, fundamental examples of vector spaces, elementary consequences of the definition, subspaces, intersections of subspaces, linear combinations of vectors, the space spanned by a set of vectors, linear independence, basis, dimension, the exchange process
	www.mcs.le.ac.uk /Modules/MA-03-04/MA1152.html (478 words)

	Vector Space
		To be able to view the math in this document, use the PDF version, or please consider using another browser, such as Mozilla, Netscape 7 or above or Microsoft Internet Explorer 6 or above (MathPlayer required for IE).
		From now on, we will denote the arbitrary vector space (V, , +, ⋅) by the shorthand V and assume the usual selection of (, +, ⋅).
		The span of set S⊂V is the subspace of V containing all linear combinations of vectors in S.
	cnx.rice.edu /content/m10419/latest (278 words)

	A brief survey of linear algebra
		The elements of the field F are called the scalars, or coefficients of the vector space.
		A useful way to think about this is that a set S of vectors is linearly independent if no individual one of the vectors is linearly dependent on a finite number of the rest of them.
		Thus, two isomorphic vector spaces are indistinguishable as vector spaces except for a renaming of the elements.
	astarte.csustan.edu /~tom/SFI-CSSS/linear-algebra/lin-alg.html (4998 words)

	course_descriptions (Site not responding. Last check: 2007-11-04)
		After introducing metric and general topological spaces, the emphasis will be on the algebraic topology of manifolds and cell complexes.
		Two examples are interfaces separating two different fluids in a multiphase flow, and between solid and liquid regions in a solidification process.
		Examples will be drawn from various neural contexts, including visual and auditory systems, decision-making, motor control, and learning and memory.
	www.math.nyu.edu /degree/course_descriptions.html (7943 words)

	Teaching and Learning Committee - July 16, 1997 (Site not responding. Last check: 2007-11-04)
		There would only be time to present real, n-dimensional vector spaces, and even these would be studied in cursory fashion.
		It was felt that there was not enough time for a comprehensive course on matrices and vector spaces, though many of the topics in the current linear algebra sequence would be covered, albeit in a hurried fashion.
		Even in a one-and-a-half course sequence, little time would be available for the study of abstract vector spaces.
	www.math.uwaterloo.ca /Ugrad_Office/TL/TL97Jul16.html (614 words)

	ECE 601 Homepage (Site not responding. Last check: 2007-11-04)
		Examples of Vector Spaces: Many different types of mathematical objects can be used to form a vector space.
		For each set described below, try to impose a vector space structure on it by explicitly defining a notion of vector addition and scalar multiplication.
		Whenever a vector space structure is not possible, explain why not.
	www.lions.odu.edu /~sgray/classes/ece601/hw1.html (248 words)

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