A rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points.
The rigid transformations include rotations, translations, reflections, or any sequence of these. A reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.
By Cartan–Dieudonné theorem in three-dimensional Euclidean space, every orthogonal transformation can be described as a single reflection, a rotation (2 reflections), or an improper rotation (3 reflections). The set of all (proper and improper) rigid transformations is a mathematical group called the Euclidean group, denoted $E(n)$ for n-dimensional Euclidean spaces. Now suppose $l$ is an arbitrary line in Euclidean three-dimensional space. Suppose that the transformation $T$ is a reflection with respect to the line $l$ in three-dimensional space. Call the group generated by all these $T$'s $G$. Can group $G$ be identified?
I assume any line here crosses the origin, otherwise it will not be a linear subspace
If y is the reflection of x with respect a plane H, then it is at the same time the reflection of x with respect to the line that connects the origin and the intersection of H and the line segment x-y, therefore the group generated by reflection of lines is at least as large as the group generated by reflection of planes, i.e., $E(n)$
On the other hand, it is clear that x,y haves equal length, so any such transformation must be rigid, i.e, it is no larger than $E(n)$, consequently it is $E(n)$
Remark: If P is the orthogonal projection onto a line, then the reflection with respect to this line is simply $2P-I$ where I is the identity.