linear manifold

Definition

Suppose $V$ is a vector space, $L$ is a non-empty subset of $V$. If there exists $v \in V$ such that $L + v = \{v+l | l \in L\}$ is a vector subspace of $V$, then $L$ is a linear manifold of $V$

Say the dimension of $L$ i s the dimension of $L+v$ i.e. $\dim L = \dim(L+v)$

Notes

• in the case $\dim L = \dim V-1$, $L$ is called a hyperplane
• linear manifold in other words is a linear subspace possibly shifted away from the origin
  • in $\mathbb{R}^2$, examples are points, lines (hyperplanes), and $\mathbb{R}^2$ itself
• there are some inconsistent definitions in functional analysis and geometry contexts between different authors, so check terminology
  • e.g. alternatively defined only for Hilbert spaces (a type of vector space) specifically, requiring closure under addition of vectors and scalar multiplication (i.e. linear subspace)

See also

• subspace

References

1. https://math.stackexchange.com/questions/1613939/what-is-the-difference-between-linear-manifold-and-linear-vector-subspace
2. https://planetmath.org/linearmanifold