Editing Orthogonal complement (section)

=== Finite dimensions ===

For a finite-dimensional inner product space of dimension <math>n</math>, the orthogonal complement of a <math>k</math>-dimensional subspace is an <math>(n-k)</math>-dimensional subspace, and the double orthogonal complement is the original subspace:
<math display="block">\left(W^{\bot}\right)^{\bot} = W.</math>

If <math>\mathbf{A} \in \mathbb{M}_{mn}</math>, where <math>\mathcal{R}(\mathbf{A})</math>, <math>\mathcal{C} (\mathbf{A})</math>, and <math>\mathcal{N} (\mathbf{A})</math> refer to the [[row space]], [[column space]], and [[null space]] of <math>\mathbf{A}</math> (respectively), then<ref>[https://www.mathwizurd.com/linalg/2018/12/10/orthogonal-complement "Orthogonal Complement"]</ref>
<math display="block">\left(\mathcal{R} (\mathbf{A}) \right)^{\bot} = \mathcal{N} (\mathbf{A}) \qquad \text{ and } \qquad \left(\mathcal{C} (\mathbf{A}) \right)^{\bot} = \mathcal{N} (\mathbf{A}^{\operatorname{T}}).</math>