In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. It is sometimes...

means that, for two vector spaces over a given field and with the same dimension, the properties that depend only on the vector-space structure are exactly...

is the manifold's dimension. For connected differentiable manifolds, the dimension is also the dimension of the tangent vector space at any point. In geometric...

with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Basis vectors find applications...

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space...

page lists some examples of vector spaces. See vector space for the definitions of terms used on this page. See also: dimension, basis. Notation. Let F denote...

are one-dimensional spaces but are usually referred to by more specific terms. Any field K {\displaystyle K} is a one-dimensional vector space over itself...

The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented...

In mathematics, the dimension theorem for vector spaces states that all bases of a vector space have equally many elements. This number of elements may...

projective space of dimension n is defined as the set of the vector lines (that is, vector subspaces of dimension one) in a vector space V of dimension n + 1...

space. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional...

In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined. A norm is a generalization...

the vector space. The dimension of an affine space is defined as the dimension of the vector space of its translations. An affine space of dimension one...

In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (coordinates)...

manifold that has the same dimension as a variety and as a manifold. If V is a variety, the dimension of the tangent vector space at any non singular point...

A topological vector space is a vector space that is also a topological space with the property that the vector space operations (vector addition and scalar...

on the above sorts of vectors. A vector space formed by geometric vectors is called a Euclidean vector space, and a vector space formed by tuples is called...

A two-dimensional space is a mathematical space with two dimensions, meaning points have two degrees of freedom: their locations can be locally described...

In mathematics, a symplectic vector space is a vector space V {\displaystyle V} over a field F {\displaystyle F} (for example the real numbers R {\displaystyle...

transformation, vector space homomorphism, or in some contexts linear function) is a mapping V → W {\displaystyle V\to W} between two vector spaces that preserves...

vector space V {\displaystyle V} by a subspace N {\displaystyle N} is a vector space obtained by "collapsing" N {\displaystyle N} to zero. The space obtained...

significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E {\displaystyle E} ), and is denoted by...

mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an...

functional analysis, a Banach space (pronounced [ˈbanax]) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows...

In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space R n {\displaystyle...

vector fields, primarily in three-dimensional Euclidean space, R 3 . {\displaystyle \mathbb {R} ^{3}.} The term vector calculus is sometimes used as a synonym...

of a coordinate vector can also be used for infinite-dimensional vector spaces, as addressed below. Let V be a vector space of dimension n over a field...

in n-dimensional space. When n = 7, the set of all such locations is called 7-dimensional space. Often such a space is studied as a vector space, without...

space and time in the coordinate form in a four-dimensional real vector space. Points in this space correspond to events in spacetime. In this space,...

six-dimensional Euclidean space, R 6 {\displaystyle \mathbb {R} ^{6}} , is generated by considering all real 6-tuples as 6-vectors in this space. As such...