A one-dimensional space (1D space) is a mathematical space in which location can be specified with a single coordinate. An example is the number line, each point of which is described by a single real number.
Any straight line or smooth curve is a one-dimensional space, regardless of the dimension of the ambient space in which the line or curve is embedded. Examples include the circle on a plane, or a parametric space curve.
In physical space, a 1D subspace is called a "linear dimension" (rectilinear or curvilinear), with units of length (e.g., metre).
In algebraic geometry there are several structures that 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 projective line over
K
,
{\displaystyle K,}
denoted
P
1
(
K
)
,
{\displaystyle \mathbf {P} ^{1}(K),}
is a one-dimensional space. In particular, if the field is the complex numbers
C
,
{\displaystyle \mathbb {C} ,}
then the complex projective line
P
1
(
C
)
{\displaystyle \mathbf {P} ^{1}(\mathbb {C} )}
is one-dimensional with respect to
C
{\displaystyle \mathbb {C} }
(but is sometimes called the Riemann sphere, as it is a model of the sphere, two-dimensional with respect to real-number coordinates).
For every eigenvector of a linear transformation T on a vector space V, there is a one-dimensional space A ⊂ V generated by the eigenvector such that T(A) = A, that is, A is an invariant set under the action of T.
In Lie theory, a one-dimensional subspace of a Lie algebra is mapped to a one-parameter group under the Lie group–Lie algebra correspondence.
More generally, a ring is a length-one module over itself. Similarly, the projective line over a ring is a one-dimensional space over the ring. In case the ring is an algebra over a field, these spaces are one-dimensional with respect to the algebra, even if the algebra is of higher dimensionality.
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Dimensions - Geometry - Types, Applications, and Examples
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