Nullspace

The nullspace of $$A \in \mathbb{R}^{m\times n}$$ is defined as

$$\mathcal{N}(A) = \{x \in \mathbb{R}^n : Ax = 0\}.$$

$$\mathcal{N}(A)$$ is the set of vectors orthogonal to all rows of $$A$$.

$$\mathcal{N}(A)$$ gives the *ambiguity* in $$x$$ given $$y=Ax$$:
 * If $$y = Ax$$ and $$z \in \mathcal{N}(A)$$, then $$y = A(x+z) = Ax$$
 * Conversely, if $$y = Ax$$ and $$y = A\tilde{x}$$, then $$\tilde{x} = x+z$$ for some $$z \in \mathcal{N}(A)$$