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...

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

finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. A basis B of a vector space V over...

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...

by more specific terms. Any field K {\displaystyle K} is a one-dimensional vector space over itself. The projective line over K , {\displaystyle K,} denoted...

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...

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

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...

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 spaces are isomorphic). A vector space is finite-dimensional if its dimension is a natural number. Otherwise, it is infinite-dimensional, and...

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...

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...

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 symplectic vector space is a vector space V over a field F (for example the real numbers R) equipped with a symplectic bilinear form...

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...

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

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...

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...

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

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,...

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...

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 = 8, the set of all such locations is called 8-dimensional space. Often such spaces are studied as vector spaces, without...

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...

multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector (that is, the dimension of the domain...

product V ⊗ W {\displaystyle V\otimes W} of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V × W...

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...

infinite-dimensional vector function is a function whose values lie in an infinite-dimensional topological vector space, such as a Hilbert space or a Banach...