orthogonal complement

Orthogonal complement

The orthogonal complement is the set of all vectors $W^{\perp }$ that are orthogonal to every vector in $W$.

$$W^{\perp }=\{v\in V\mid v\perp w,\forall w\in W\}$$

For a finite-dimensional inner product space, $(W^{\perp }{)}^{\perp }=W$.
And for a matrix $A$ with row space $\mathcal{R}(A)$, column space $\mathcal{C}(A)$, and null space $\ker (A)$:

$$\begin{array}{rl}\mathcal{R}(A{)}^{\perp }& =\ker (A)\\ \mathcal{C}(A{)}^{\perp }& =\ker (A^T)\end{array}$$