Vector space From Wikipedia, the free encyclopedia Jump to: navigation, search This article is about linear (vector) spaces. For the structure in incidence geometry, see Linear space (geometry). Vector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2·w.
A vector space is a mathematical structure formed by a collection of elements called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below. An example of a vector space is that of Euclidean vectors, which may be used to represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples; one should think of these vectors as abstract mathematical objects which hold specific properties and in some cases, they can be visualized as arrows.
Vector spaces are the subject of linear algebra and are well understood from this point of view, since vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. A vector space may be endowed with additional structure, such as a norm or inner product. Such spaces arise naturally in mathematical analysis, mainly in the guise of infinite-dimensional function spaces whose vectors are functions. Analytical problems call for the ability to decide whether a sequence of vectors converges to a given vector. This is accomplished by considering vector spaces with additional structure, mostly spaces endowed with a suitable topology, thus allowing the consideration of proximity and continuity issues. These topological vector spaces, in particular Banach spaces and Hilbert spaces, have a richer theory.
Historically, the first ideas leading to vector spaces can be traced back as far as 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors. The modern, more abstract treatment, first formulated by Giuseppe Peano in the late 19th century, encompasses more general objects than Euclidean space, but much of the theory can be seen as an extension of classical geometric ideas like lines, planes and their higher-dimensional analogs.
Today, vector spaces are applied throughout mathematics, science and engineering. They are the appropriate linear-algebraic notion to deal with systems of linear equations; offer a framework for Fourier expansion, which is employed in image compression routines; or provide an environment that can be used for solution techniques for partial differential equations. Furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra.
Algebraic structures Group-like structures[show] Ring-like structures[show] Lattice-like structures[show] Module-like structures[hide] Group with operators
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Vector space Algebra-like structures[show] v t e Contents 1 Introduction and definition 1.1 First example: arrows in the plane 1.2 Second example: ordered pairs of numbers 1.3 Definition 1.4 Alternative formulations and elementary consequences 2 History 3 Examples 3.1 Coordinate spaces 3.2 The complex numbers and other field extensions 3.3 Function spaces 3.4 Linear equations 4 Bases and dimension 5 Linear maps and matrices 5.1 Matrices 5.2 Eigenvalues and eigenvectors 6 Basic constructions 6.1 Subspaces and quotient spaces 6.2 Direct product and direct sum 6.3 Tensor product 7 Vector spaces with additional structure 7.1 Normed vector spaces and inner product spaces 7.2 Topological vector spaces 7.2.1 Banach spaces 7.2.2 Hilbert spaces 7.3 Algebras over fields 8 Applications 8.1 Distributions 8.2 Fourier analysis 8.3 Differential geometry 9 Generalizations 9.1 Vector bundles 9.2 Modules 9.3 Affine and projective spaces 9.4 Convex analysis 10 See also 11 Notes 12 Footnotes 13 References 13.1 Linear algebra 13.2 Analysis 13.3 Historical references 13.4 Further references 14 External links Introduction and definition First example: arrows in the plane The concept of vector space will first be explained by describing two particular examples. The first example of a vector space consists of arrows in a fixed plane, starting at one fixed point. This is used in physics to describe forces or velocities. Given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted v + w. Another operation that can be done with arrows is scaling: given any positive real number a, the arrow that has the same direction as v, but is dilated or shrunk by multiplying its length by a, is called multiplication of v by a. It is denoted av. When a is negative, av is defined as the arrow pointing in the opposite direction, instead.
