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 necessary have to be an 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.
Semigroup and Monoid
Quasigroup and Loop
Map of lattices
Group with operators
Other articles related to "vector space, space, spaces, vectors, vector spaces":
... A Lie algebra structure on a vector space is a map which is skew-symmetric, and satisfies the Jacobi identity ... Dually, a Lie coalgebra structure on a vector space E is a linear map which is antisymmetric (this means that it satisfies, where is the canonical flip ) and satisfies the so-called cocycle condition (also known as ... The dual space E* carries the structure of a bracket defined by α = dα(x∧y), for all α ∈ E and x,y ∈ E* ...
... as the combinations of the axioms for an affine space and a cone, which is reflected in the standard space for it, the n-simplex, being the intersection of the affine hyperplane ... Such spaces are particularly used in linear programming ... In the language of universal algebra, a vector space is an algebra over the universal vector space K∞ of finite sequences of coefficients, corresponding to finite sums of vectors, while an affine space is an ...
... In mathematics, a prehomogeneous vector space (PVS) is a finite-dimensional vector space V together with a subgroup G of GL(V) such that G has an open dense orbit in V ... Prehomogeneous vector spaces were introduced by Mikio Sato in 1970 and have many applications in geometry, number theory and analysis, as well as representation theory ...
... Dimension of a vector space V over a field K is the cardinality of any Hamel basis of V ... For a linear subspace W of a vector space V we define codimension of W (with respect to V) ... (note that the algebraic degree equals the dimension of the extension as a vector space over the smaller field) ...
... In the first diagram the circled nodes represent 1-spaces and 2-spaces in a three-dimensional vector space ... In the second diagram the circled nodes represent 1-space and 2-spaces in a 3-dimensional vector space over the quaternions, which in turn represent certain 2-spaces and 4-spaces in a 6-dimensional complex vector space ...
Famous quotes containing the word space:
“Not so many years ago there there was no simpler or more intelligible notion than that of going on a journey. Travelmovement through spaceprovided the universal metaphor for change.... One of the subtle confusionsperhaps one of the secret terrorsof modern life is that we have lost this refuge. No longer do we move through space as we once did.”
—Daniel J. Boorstin (b. 1914)