Properties of Linear Subspace of Finite Dimensional Scalar Product Space

Theorem
Let $\struct {V, q}$ be a scalar product space.

Let $S \subseteq V$ be a linear subspace.

Then:


 * $\dim S + \dim S^\perp = \dim V$


 * $\paren {S^\perp}^\perp = S$

where $\dim$ denotes the dimension of vector space, and $S^\perp$ denotes the vector subspace perpendicular to $S$ with respect to $q$.