Definition Suppose $V$ is a vector space and suppose that $L$ is a non-empty subset of $V$. If there exists a $v\in V$ such that $L+v=\{v+l\mid l\in L\}$ is a vector subspace of $V$, then $L$ is a linear manifold of $V$. Then we say that the dimension of $L$ is the dimension of $L+v$ and write $\dim L=\dim(L+v)$. In the important case $\dim L=\dim V-1$, $L$ is called a hyperplane.

A linear manifold is, in other words, a linear subspace that has possibly been shifted away from the origin. For instance, in $\mathbb{R}^{2}$ examples of linear manifolds are points, lines (which are hyperplanes), and $\mathbb{R}^{2}$ itself. In $\mathbb{R}^{n}$ hyperplanes naturally describe tangent planes to a smooth hyper surface.