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Topic: Null vector (vector space)

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	Encyclopedia :: encyclopedia : Vector space (Site not responding. Last check: 2007-10-11)
		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.
	www.hallencyclopedia.com /topic/Vector_space.html (1066 words)

	Null vector (vector space) - Wikipedia, the free encyclopedia
		In linear algebra and related areas of mathematics, the null vector or zero vector is the vector (0, 0, …, 0) in Euclidean space, all of whose components are zero.
		For a general vector space, the null vector is the uniquely determined vector that is the identity element for vector addition.
		The preimage of the zero vector under a linear transformation f is called kernel or null space.
	en.wikipedia.org /wiki/Null_vector_(vector_space) (174 words)

	PlanetMath: geometric algebra
		Any vector in this space may be represented by a linear combination of the basis vectors.
		The parallelism of vectors is encoded as a symmetric property, while orthogonality of vectors is encoded as an antisymmetric property.
		The contraction rule specifies that the square of any vector is a scalar equal to the sum of the square of the magnitudes of its components in each basis direction.
	planetmath.org /encyclopedia/GeometricAlgebra.html (493 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 zero vector is always in the null space and the null space is a subspace of the domain space.
		The row space of a matrix is the subspace of the domain which is spanned by the rows of the matrix.
	www.math.umbc.edu /~campbell/Math221/Glossary (1250 words)

	Vector (spatial) Summary
		The vector representing the plane's intended velocity points due east and is labeled 600 mph, while the vector representing the wind velocity points from the northeast at a 45° angle to the line heading due east.
		A spatial 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).
		Vectors can be contrasted with scalar quantities such as distance, speed, energy, time, temperature, charge, power, work, and mass, which have magnitude, but no direction (they are invariant under coordinate rotations).
	www.bookrags.com /Vector_(spatial) (5261 words)

	Minkowski space Summary
		There is an alternative definition of Minkowski space as an affine space which views Minkowski space as a homogeneous space of the Poincaré group with the Lorentz group as the stabilizer.
		Vector fields are called timelike, spacelike or null if the associated vectors are timelike, spacelike or null at each point where the field is defined.
		Minkowski space is named for the German mathematician Hermann Minkowski, who around 1907 realized that the theory of special relativity previously worked out by Einstein and Lorentz could be elegantly described using a four-dimensional spacetime, which combines the dimension of time with the three dimensions of space.
	www.bookrags.com /Minkowski_space (1466 words)

	Notes for Exam 3 (Site not responding. Last check: 2007-10-11)
		A vector space V is a set of objects, called "vectors," with two operations: vector addition and scalar multiplication.
		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
	www.math.byu.edu /~dsiebert/math343/prevnotes3.html (3141 words)

	Linear Algebra
		Mathematicians have generalized the definition of a vector space: a general vector space has the properties we’ve listed above for three-dimensional real vectors, but the operations of addition and multiplication by a number are generalized to more abstract operations between more general entities.
		To go from the familiar three-dimensional vector space to the vector spaces relevant to quantum mechanics, first the real numbers (components of the vector and possible multiplying factors) are to be generalized to complex numbers, and second the three-component vector goes an
		This notion naturally extends to vectors and numbers: the adjoint of a ket is the corresponding bra, the adjoint of a number is its complex conjugate.
	galileo.phys.virginia.edu /classes/751.mf1i.fall02/751LinearAlgebra.htm (2711 words)

	java.util.Vector
		A vector is similar to an array of objects except that when the specified capacity is exceeded,the Vector automatically grows by a specified increment amount.
		Vectors are stored in memory wherever space is available, therefore the elements that are virtually next to each other may be stored far away in memory.
		Adds the specified element to this Vector at the given index (which may be no bigger than the current size) Other elements at and after this index are shifted upward to make room for the new element.
	www2.hawaii.edu /~blanca/spring2002/notes/0122-Vectors.htm (1277 words)

	Linear Dependence of Vectors. Basis of the Vector Space.
		Prove that each set of vectors that contains the null vector is linearly dependent.
		A subset B of the vectors of the vector space
		is a basis of the space iff it is the maximal linearly independent subset.
	www.cs.ut.ee /~toomas_l/linalg/lin1/node7.html (377 words)

	ESS265: History of Vector Magnetometry:
		The sensor should truly make vector measurements as the angular response should be equal to B cos f where f is the angle between the field vector and the sensor axis.
		One sensor was parallel to the spacecraft spin axis one was normal to the spin and the third was at a point 2/3 of the length of the 5m boom and tilted at a 45o angle in the radial direction.
		At low frequencies in quiet fields, calculation of the vector field from the sine wave amplitude and phase and the steady component parallel to the spin axis could be provided by either the inboard tilted sensor on the two outboard sensors.
	www-ssc.igpp.ucla.edu /personnel/russell/ESS265/History.html (6585 words)

	Mathematical Preliminaries
		Vectors are usually thought of as living in a vector space--the set of all possible values that such a vector could take on.
		A vector space is closed: by taking a linear combination of vectors in the space you always get another vector in the space.
		The columns of the first matrix are vectors to be transformed (expressed in the old coordinate system) and the rows of the second matrix are the new coordinate basis vectors (also expressed in the old coordinate system).
	rivit.cs.byu.edu /morse/550-F95/node9.html (2071 words)

	Class Summaries( 8-Apr-1997) (Site not responding. Last check: 2007-10-11)
		Another way to see this is that the dimension of the domain space (no. of columns) is the sum of the dimension of the null space and the dimension of its range.
		A linear transformation between abstract vector spaces V and W can be written in matrix form once ordered bases have been chosen for V and W. A choice of bases, depending on T, can be chosen so that the matrix has all zero entries except for possibly on the diagonal.
		A steady state vector is an eigen vector with eigenvalue one for the transition matrix.
	www.math.uchicago.edu /~todd/lclass/class.html (522 words)

	Ket space
		Thus, if we were to represent a classical wave by a vector, then the magnitude, or length, of the vector would correspond to the amplitude of the wave, and the direction would correspond to the frequency and wave-length, so that two vectors of different lengths pointing in the same direction would represent different wave states.
		We conclude that a ket space must be a complex vector space if it is to properly represent the mutual interrelations between the possible states of a microscopic system.
		The dimensionality of a conventional vector space is defined as the number of independent vectors contained in the space.
	farside.ph.utexas.edu /teaching/qm/lectures/node7.html (994 words)

	[No title]
		If you want to find a collection of integral vectors with the property that any integral vector in the null space is an _integral_ linear combination of the ones in the first set, then you are asking a question about free modules over the integers.
		I'm not sure that adding the equation that forces these other null vectors to be perpendicular to the first (for example) is anywhere close to the optimal situation.
		The norm of this vector (usual L2 norm) is the "height" of the rowspace.
	www.math.niu.edu /~rusin/known-math/95/null (1862 words)

	A Unified Algebraic Framework for Classical Geometry
		This deficiency in the vector space model was corrected early in the 19th century by removing the origin from the plane and placing it one dimension higher.
		If one of these points is a null vector e representing the point at infinity, then a \wedge b \wedge e represents the straight line through a and b as a circle through infinity.
		Because the three geometries are obtained by interpreting null vectors of the same Minkowski space differently, natural correspondences exist among geometric entities and constraints of these geometries.
	modelingnts.la.asu.edu /html/UAFCG.html (2056 words)

	Algorithms and Special Topics
		Direct conversion from vector shorts and vector chars is possible with this general method, and in many cases can be faster than the more standard method.
		The central problem facing would-be vectorizers for this task, is that using the traditional algorithm, rows of matrix A are multiplied by columns of matrix B, which may mean transposing matrix B. In addition you then have to sum across the result, which is not very efficient.
		Commonly when working with constant terms in vector code, it is desirable to be able to load a scalar and splat it across a register.
	developer.apple.com /hardware/ve/algorithms.html (3278 words)

	Math 307 §B (Site not responding. Last check: 2007-10-11)
		Null space, column space and row space of a matrix.
		Basis of a vector space or vector subspace.
		Coordinates of a vector relative to a basis.
	www.public.iastate.edu /~boushaba/index1.htm (184 words)

	Vector Spaces
		One of the fundamental concepts of linear algebra is the concept of vector space.
		In analysis the notion ``linear space'' is used unstead of the notion ``vector space''.
		Instead of the notion ``vector space'' we shall use the abbreviative ``space''.
	www.cs.ut.ee /~toomas_l/linalg/lin1/node5.html (382 words)

	Math 307 (Site not responding. Last check: 2007-10-11)
		The vector (0,b_2,b_3) ranges through all of the vectors in the plane, as the variables b_2 and b_3 range through the reals.
		Hint 3: This subspace is the Null space of a 1 by 3 matrix A (that is, it is the solution space for a single homogeneous linear equation).
		Note that if you know X is a subspace (of any vector space) then you know X is a vector space.
	www.math.ubc.ca /~jessica/math307/email.html (865 words)

	Review of Chapters 5 and 6 and loose ends
		The dimension of a vector space is the number of vectors in a basis for the space.
		The rank of a matrix is equal to the dimension of the column space, which is also equal to the dimension of the row space.
		Nice fact: The row space of A is the orthogonal complement of the null space of A (and vice versa).
	www.math.uiuc.edu /~johnmaki/ch6review.html (1579 words)

	Nichols: Null space
		The properties of the null space can be explored by examining the singular value decomposition of the operator.
		This operator has two model space vectors one of which is in the null space.
		The second vector with a singular value of zero, and thus in the null space is
	sepwww.stanford.edu /public/docs/sep82/dave1/paper_html/index.html (1149 words)

	Bra space
		This type of vector space is called a bra space (after Dirac), and its constituent vectors (which are actually functionals of the ket space) are called bra vectors.
		Recall that a bra vector is a functional which acts on a general ket vector, and spits out a complex number.
		, and is analogous to the length, or magnitude, of a conventional vector.
	farside.ph.utexas.edu /teaching/qm/lectures/node8.html (567 words)

	Math 225, Review for the Final Exam (Site not responding. Last check: 2007-10-11)
		Theorems 4.1, 4.2, 4.3 (span, null space and column space are subspaces)
		Theorem 6.3 (null space of A is orthogonal to the row space of A)
		Determine when a vector is in a subset spanned by specified vectors.
	www.math.uiuc.edu /~dcmurphy/math225/reviewf.html (810 words)

	Kerr Black Holes, Ch.5. Petrov Types
		The complex operator IC determines real null directions at each point of M, and the resulting principal null congruences lead to a second characterization of Petrov type.
		To study the geometry of null congruences a natural weapon is the Newman-Penrose formalism, in which orthonormal frame fields are replaced by frame fields containing null vector fields.
		In particular, for type D spacetimes such as Kerr's, these congruences are not only geodesic but also "shearfree"--roughly speaking, if a beam of such light initially has circular cross-section, then it keeps this property as it propagates.
	www.math.ucla.edu /~bon/kerr/intro5.html (336 words)

	VML - the Vector Markup Language
		VML is an application of Extensible Markup Language (XML) 1.0 which defines a format for the encoding of vector information together with additional markup to describe how that information may be displayed and edited.
		A segment of the ellipse is drawn which starts at the angle defined by the start radius vector and ends at the angle defined by the end vector.
		Dash styles are defined in terms of the length of the dash (the drawn part of the stroke) and the length of the space between the dashes.
	www.w3.org /TR/1998/NOTE-VML-19980513 (8535 words)

	Math 2307_10962
		A basis S for a vector space V must have enough vectors to span V but not so many that one of them could be written as a linear combination of the other vectors in S. See page 215.
		Section 4.3 A nonempty subset W of a vector space V is a subspace if for any vectors u and v in W, the sum u+v is in W, and if for every scalar c, the product cu is in W. Keep reviewing the fundamentals of vector spaces and subspaces.
		A subspace is a vector space that is a subset of another vector space.
	cms.dt.uh.edu /Faculty/BecerraL/Fall2005/2307_calendar.htm (2449 words)

	[No title]
		* * Nullspace() computes the vectors x of order n such that xa = z, * where a is an nxn matrix, z is the zero-vector of order n, * eps is a small positive number such that if the maximum pivot * element is numerically less than eps the procedure considers it * zero.
		The ec vectors x are to be found in the first ec rows * of the matrix a upon exit from this procedure.
		The size of the * null space, ec, is the return value from this procedure.
	www.worldserver.com /turk/opensource/NullSpace.c.txt (526 words)

	Null Space
		Summary: This module defines the null space, shows an example of what one is, and describes how to find one given an arbitrary matrix.
		There is a great deal more to finding null spaces; enough, in fact, to warrant another module.
		One important aspect and use of null spaces is their ability to inform us about the uniqueness of solutions.
	cnx.org /content/m10293/latest (516 words)

	PlanetMath: kernel of a linear transformation
		See Also: linear transformation, image of a linear transformation, nullity, rank-nullity theorem
		Cross-references: matrices, transformations, invertible, finite, injective, nullity, vector subspace, maps, vectors, vector spaces, linear transformation
		This is version 14 of kernel of a linear transformation, born on 2001-11-13, modified 2005-12-01.
	planetmath.org /encyclopedia/Nullspace.html (103 words)

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