Dimensions of Linear Objects

Embedding Dimension and Intrinsic Dimension

The embedding dimension of a geometric object is the dimension of the ambient space in which it resides. The intrinsic dimension is the minimum number of independent parameters (or coordinates) required to describe a point within that object.

For example, consider a set consisting of a single point $\{a\}$ where $a\in R^n$. Its embedding dimension is $n$ because it resides within an $n$-dimensional ambient space. Its intrinsic dimension is $0$ because the set contains only a fixed location, and therefore zero parameters are required to describe a point within it.

Dimensionality Table of Linear Objects

Name	Embedding dim.	Intrinsic dim.
Point	n	0
Line	n	1
Plane	n	2
Hyperplane	n	n-1
Space	n	n