Dimension (mathematics)
Introduction
In Mathematics, a dimension is a fundamental concept that describes the number of degrees of freedom available in a mathematical space. The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathematics and the sciences. They may be abstract spaces, independent of the physical space we live in.
Definition
The dimension of a vector space is the number of vectors in any basis for the space, i.e. the number of coordinates necessary to specify any vector. This notion of dimension (the cardinality of a basis) is often referred to as the Hamel dimension or algebraic dimension to distinguish it from other definitions.
Dimension in Geometry
In Geometry, the dimension of an object is an intrinsic property, independent of the space in which the object may happen to be embedded. For example, a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate (the polar coordinate angle), so the circle is 1-dimensional even though it exists in the 2-dimensional plane.
Dimension in Topology
In Topology, dimension can be defined in multiple ways. The most intuitive way is through covering dimension. A base B for a topological space X with the property that every open cover of X has a refinement base which is locally finite, is said to have dimension n if n is the smallest integer for which the following holds: any finite open cover of X has an open refinement such that no point is included in more than n+1 elements.
Dimension in Physics
In Physics, the dimension of a physical quantity is the exponent (or power) to which each of the base quantities is raised to represent that quantity. For example, speed has the dimension length/time, and the dimension of force is mass×length/time².
Fractal Dimension
Fractals are shapes that are self-similar at all scales. The concept of dimension can be extended to them using the idea of a fractal dimension, which can take non-integer values. This is a measure of the space-filling capacity of a pattern that tells how a fractal scales differently than the space it is embedded in.
Dimension in Algebraic Geometry
In Algebraic Geometry, the dimension of an algebraic variety V in algebraically closed field k is defined to be the transcendence degree of the field extension k(V)/k.
Dimension in Linear Algebra
In Linear Algebra, 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 called the Hamel dimension or the algebraic dimension to distinguish it from other types of dimension.
Dimension in Set Theory
In Set Theory, the dimension of a set is defined as the minimum number of coordinates needed to specify each point within it. This concept is used in the study of fractals and other complex geometric structures.