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	PlanetMath: vector
		Indeed, it is customary to depict a physical vector as an arrow.
		An abstract vector is an element of a vector space.
		Essentially, given a finite dimensional abstract vector space, a choice of a coordinate frame (which is really the same thing as a basis) sets up a linear bijection between the abstract vectors and list vectors, and makes it possible to represent the one in terms of the other.
	planetmath.org /encyclopedia/Vector2.html (1189 words)

	Vector space - Wikipedia, the free encyclopedia
		A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra.
		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.
		A vector space with a topology compatible with the operations (i.e., such that addition and scalar multiplication are continuous maps) is called a topological vector space.
	en.wikipedia.org /wiki/Vector_space (1072 words)

	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.
		A vector space with a defined length concept, i.e., a norm, is called a normed vector space.
		The most basic physical vector is the displacement vector from point A to point B (its direction is from A to B and its length is the distance between A and B).
	open-encyclopedia.com /Vector_space (1073 words)

	Vector space Summary
		Vector spaces are the basic objects of study in linear algebra, and are used throughout mathematics, the sciences, and engineering.
		Vectors in these spaces are ordered pairs or triples of real numbers, and are often represented as geometric vectors (quantities with a magnitude and a direction, usually depicted as arrows).
		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.
	www.bookrags.com /Vector_space (4366 words)

	PlanetMath: vector space
		Cross-references: vectors, ring, module, operations, division ring, field
		This is version 12 of vector space, born on 2001-10-19, modified 2006-07-22.
		Hungerford's _Algebra_ defines vector spaces as unitary modules over division rings, which means funky, noncommutative things can happen (you can have a "left" vector space which isn't a "right" vector space, for example).
	planetmath.org /encyclopedia/VectorSpace.html (290 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.
		The "vectors" need not be actually geometric vectors, but can be any mathematical object that satisfies the following vector space axioms.
	encyclopedia.worldvillage.com /s/b/Vector_space (975 words)

	A Simple Guide to tvector
		A vector is more like a DVD than a video cassette: you can access the 20th track of a DVD without fast-forwarding past all the tracks 1-19, but you can't do this with a cassette.
		In our first uses of vector, we'll also specify how many elements a vector can hold and what the initial values of these elements are.
		If more space is needed in the vector to add a new element, more space is created in an efficient manner (see Chapter 8 in Tapestry for details).
	www.cs.duke.edu /csed/tapestry/tvector.html (1087 words)

	[No title]
		Examples of vector quantities are: velocity, acceleration, force, electric field, magnetic field etc and will be denoted by v, a, F, E, B, etc. Two vectors a and b are equal if they have the same magnitude and direction irrespective of their initial points.
		Example — Magnetic Field The force F experienced by a point charge q moving with velocity v in a magnetic field of flux density B is given by F = q v x B Example — Rotation Consider a rigid body rotating with angular speed w about an axis.
		Examples The vectors a = 3i + 5j — 2k, b = 4j + 2k and c = i + j — k are linearly dependent since EMBED Equation.3 Hence vector a lies in the plane of the vectors b and c.
	personal.cityu.edu.hk /~mawoo/VectorAlgebra.doc (2369 words)

	Maths - Vectors - Martin Baker
		Vectors are strongly related to matrices, they can be considered as a one directional matrix, or conversely, we could construct a matrix from a vector (drawn as a column) whose elements are themselves vectors (drawn as a row) :
		Regardless of whether we consider vectors as a special case of matrices, or matrices as vectors of vectors, or if we consider vectors and matrices as different types, using vectors and matrices together is very important.
		An example of this is Einsteinean space-time, space and time dimensions square to different values, if space squares to positive then time squares to negative and visa-versa.
	www.euclideanspace.com /maths/algebra/vectors/index.htm (1816 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".
		However the set of integers is not closed under division, because dividing 3 by 2 (for example) doesn't result in a member of the set of integers.
	en.wikibooks.org /wiki/Algebra:Vector_spaces (814 words)

	Physics Help and Math Help - Physics Forums - vector space... help!!!!
		Since the definition of vector space is, as you said, a set of vectors together with two operations, this has to be assuming the "standard" operations on R^3: "coordinatewise" addition and scalar multiplication.
		In order that a subset of a vector space be a sub-space (a vector space using the same operations) whenever u and v are in the set, u+v must "also" (as well as u and v) be in that same set.
		S is not a vector space unless we have the "closure properties" for the operations.
	www.physicsforums.com /printthread.php?t=30002 (1497 words)

	1
		In this section we state the axioms of a vector space; discuss the two major types of vector spaces that will be of interest to us, n-dimensional Euclidean space and function spaces; define what is meant by a subspace, and then examine some examples of subspaces.
		Definition Let V be a vector space and W be a non-empty subset of V that is a vector space, relative to the operations of vector addition and scalar multiplication defined for V. Then W is called a subspace of V. Theorem 1.
		A 2 x 2 matrix is said to be symmetric if the entry in its first row, second column is equal to the entry in its second row, first column.
	www.math.ucla.edu /~ronmiech/33a/VecSp_SubSp/VECTORSPACE.html (3008 words)

	Vector Space
		An abstract real vector space is an additive commutative group with one additional operation: its elements may be multiplied by real numbers (scalars).
		Thus, vectors and Vector Spaces are born simultaneously.
		For example, the following pairs of 3-tuples are orthogonal: (1,0,0) and (0,1,0), (1,0,1) and (2,1,-2).
	www.cut-the-knot.org /do_you_know/mul_scal1.shtml (359 words)

	GAP Manual: 32. Vectors (Site not responding. Last check: 2007-11-04)
		A vector space V is a set of vectors, for which an addition u + v and a multiplication by scalars, i.e., elements from F, s v must be defined.
		Because vectors are just a special case of lists, all the operations and functions for lists are applicable to vectors also (see chapter Lists).
		Vectors play an important role for matrices (see chapter Matrices), which are implemented as lists of vectors.
	www.math.uiuc.edu /Software/GAP-Manual/Vectors.html (482 words)

	Subspace 1
		The similar situation may happen for one vector space contained in another in a compatible way.
		) of symmetric matrices is a subspace of the vector space
		A subspace, with the inherited operations, is a vector space.
	algebra.math.ust.hk /vector_space/02_subspace/lecture1.shtml (421 words)

	What is a State Vector?
		A state vector is a set of data telling exactly where the shuttle is in its orbit in space.
		For example, if an activity requires video downlink from the shuttle, that means that the shuttle has to have one of the two TDRSS satellites in view of its antenna.
		State vectors are as important to the ground support personnel as they are to the shuttle itself.
	science.nasa.gov /Realtime/rocket_sci/orbmech/state/state.html (564 words)

	Real Vector Spaces
		Since a vector space has a constant number of vectors in a basis, that number n is characteristic for that space and is called the dimension of that space.
		The space generated by D is called the row space of A. The rows of A are a generating set of the row space.
		A is the supplementary vector space of B with respect to V. B is the supplementary vector space of A with respect to V. A and B are supplementary vector spaces with respect to V. Basis and direct sum
	home.scarlet.be /~ping1339/vect.htm (4070 words)

	Topological Vector Space
		A topological vector space is a vector space with a topology, such that addition and scaling are continuous.
		A normed vector space is a topological vector space, deriving its topology from the metric.
		In a metric space, the translate of an open ball is an open ball, since the distance between two points does not change; but in a topological group, we have to use the properties of continuity to prove translation preserves open sets.
	www.mathreference.com /top-ban,tvs.html (1378 words)

	Calculus II (Math 2414) - Vectors - Vector Arithmetic
		Note that we can’t add or subtract two vectors unless they a have the same number of components. If they don’t have the same number of components then addition and subtraction can’t be done.
		Example 2 Determine if the sets of vectors are parallel or not.
		Okay, what we’re asking for is a new parallel vector (points in the same direction) that happens to be a unit vector. We can do this with a scalar multiplication since all scalar multiplication does is change the length of the original vector (along with possibly flipping the direction to the opposite direction).
	tutorial.math.lamar.edu /AllBrowsers/2414/VectorArithmetic.asp (643 words)

	Read This: Geometrical Vectors
		He starts from the usual vectors (conceived of as arrows), but argues that a quantity should be represented by an arrow vector only if it transforms correctly under all topological transformations of the space.
		There is clearly a way to translate such a stack vector to an arrow: take the arrow perpendicular to the family of parallel planes, of a length proportional to the density of the stack.
		To the mathematician, this is the clue: "stack vectors" are elements of the vector space dual to the space of arrows.
	www.maa.org /reviews/vectors.html (1359 words)

	On-Line Geometric Modeling Notes
		Examples are drawn from the vector space of vectors in
		In the case of the space of 2-dimensional vectors, the summation is componentwise (i.e.
		Frequently this 2-d vectors is protrayed as joining the ends of the two original vectors.
	graphics.idav.ucdavis.edu /education/CAGDNotes/Vector-Spaces/Vector-Spaces.html (639 words)

	Vector Spaces
		Thus we have a situation which may yield a vector space.
		The zero vector is the constant function (it is continuous) which is identically zero on
		Example 2 Give a example of a function which is in
	distance-ed.math.tamu.edu /Math640/chapter3/node20.html (284 words)

	Elementary Vector Analysis - HMC Calculus Tutorial
		Vectors can be defined in any number of dimensions, though we focus here only on 3-space.
		When drawing a vector in 3-space, where you position the vector is unimportant; the vector's essential properties are just its magnitude and its direction.
		The coordinate vectors are examples of unit vectors.
	www.math.hmc.edu /calculus/tutorials/vectoranalysis (547 words)

	Vector Model
		Formally, a vector space is defined by a set of linearly independent basis vectors.
		Get your vector space of terms (the vocabulary for t>e dataset) Both documents and queries (indeed all relevant objects) are represented by vectors.
		Example: atc for documents and atn for queries.
	mingo.info-science.uiowa.edu:16080 /courses/230/Lectures/Vector1.html (677 words)

	LANprint User Manual
		The blank characters, space and tab, are used to separate items and cannot normally be included in an item.
		To simplify the syntax definition expressions where it is not necessary to be explicit, blank characters are simply left as blank spaces between items and the presence of a character at the end of a line is assumed.
		Vector Networks is happy for customers to print additional copies of this manual providing they remain unmodified; Vector Networks retains full copyright over the manual content.
	www.vector-networks.com /lanprint/manual/lpp001.htm (2979 words)

	Cascading Style Sheets, level 1
		But, as the example shows, the placement of an element is relative to ancestors and siblings, so these elements' padding and margin properties have an effect on their children.
		For example, by setting the 'float' property of an image to 'left', the image is moved to the left until the margin, padding or border of another block-level element is reached.
		Examples of constraints of the presentation medium are: limited resources (fonts, color) and limited resolution (so margins may not be accurate).
	www.w3.org /TR/REC-CSS1 (15136 words)

	On-Line Computer Graphics Notes
		The simplest example of a vector space to understand is the space of vectors in the 2-dimensional plane.
		The set of vectors in 2-dimensional space forms a vector space, and so has two algebraic operations: addition and scalar multiplication.
		The axioms of a vector space are, in most cases, represented geometrically.
	graphics.cs.ucdavis.edu /education/GraphicsNotes/Vectors-in-2d/Vectors-in-2d.html (353 words)

	Subspaces of the Vector Space
		To prove sufficiency, we have to show that in our case conditions 1-8 of the vector spaces are satisfied.
		(example 1.1.3) is a subspace of vector space
		Prove that the set of all symmetric matrices form a subspace in the vector space of all square matrices
	www.cs.ut.ee /~toomas_l/linalg/lin1/node6.html (323 words)

	State Space Example
		The state space representation of a system is one way in which the mathematical model of the system can be expressed.
		The state space model for an nth-order system is a set of n 1st-order differential equations, called the state equations, and a set of p algebraic equations, called the output equations.
		As an example of how the initial conditions cause different outputs for the two models, each model was given an initial state vector consisting of 1 for each element (4 of them for the full-order model, 3 for the reduced-order model).
	bass.gmu.edu /ececourses/ece220_320_421/stspacex.html (1794 words)

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