Vector spaces

A vector space (also called a linear space) is a set of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. 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 vector axioms (listed below in § Definition). To specify that the scalars are real or complex numbers, the terms real vector space and complex vector space are often used.
Certain sets of Euclidean vectors are common examples of a vector space. They represent physical quantities such as forces, where 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 way (but in a more geometric sense), vectors representing displacements in the plane or 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: vectors are regarded as abstract mathematical objects with particular properties, which in some cases can be visualized as arrows.
Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis as function spaces, whose vectors are functions. These vector spaces are generally endowed with some additional structure such as a topology, which allows the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are more commonly used (being equipped with a notion of distance between two vectors). This is particularly the case of Banach spaces and Hilbert spaces, which are fundamental in mathematical analysis.
Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's analytic geometry, matrices, systems of linear equations and Euclidean vectors. The modern, more abstract treatment, first formulated by Giuseppe Peano in 1888, 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. They offer a framework for Fourier expansion, which is employed in image compression routines, and they 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.
This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.

View More On Wikipedia.org
  • 264

    Greg Bernhardt

    A PF Singularity From USA
    • Messages
      19,445
    • Media
      227
    • Reaction score
      10,025
    • Points
      1,237
  • 4

    gruba

    A PF Electron
    • Messages
      206
    • Reaction score
      1
    • Points
      16
  • 2

    iJake

    A PF Quark
    • Messages
      41
    • Reaction score
      0
    • Points
      4
  • 1

    Santiago24

    A PF Electron 21 From Uruguay
    • Messages
      32
    • Reaction score
      6
    • Points
      16
  • 1

    nomadreid

    A PF Mountain From Israel
    • Messages
      1,672
    • Reaction score
      206
    • Points
      212
  • 1

    Hall

    A PF Atom
    • Messages
      351
    • Reaction score
      87
    • Points
      38
  • 1

    Korybut

    A PF Molecule From Russian Federation
    • Messages
      60
    • Reaction score
      2
    • Points
      91
  • 1

    Ineedhelpimbadatphys

    A PF Electron
    • Messages
      9
    • Reaction score
      2
    • Points
      13
  • 1

    jv07cs

    A PF Quark
    • Messages
      39
    • Reaction score
      2
    • Points
      3
  • 1

    kostoglotov

    A PF Electron From Brisbane, Australia
    • Messages
      234
    • Reaction score
      6
    • Points
      20
  • 1

    "Don't panic!"

    A PF Atom
    • Messages
      601
    • Reaction score
      8
    • Points
      46
  • 1

    Prof. 27

    A PF Atom
    • Messages
      50
    • Reaction score
      1
    • Points
      31
  • 1

    Apothem

    A PF Electron
    • Messages
      39
    • Reaction score
      0
    • Points
      11
  • 1

    Austin Chang

    A PF Quark
    • Messages
      38
    • Reaction score
      0
    • Points
      1
  • 1

    Alex Langevub

    A PF Quark
    • Messages
      4
    • Reaction score
      0
    • Points
      4
  • 1

    Karl Karlsson

    A PF Electron From Stockholm
    • Messages
      104
    • Reaction score
      12
    • Points
      21
  • 1

    shinobi20

    A PF Cell
    • Messages
      267
    • Reaction score
      19
    • Points
      106
  • Back
    Top