The null space of a matrix is a subspace of R^n (or C^n).

**Theorem:** The *null space* of an \(m \times n\) matrix \(A\) is a subspace of \(\mathbb{F}^n\):
\[\text{null}(A) = \{\mathbf{x} \in \mathbb{F}^n : A\mathbf{x} = \mathbf{0}\}\]

**Proof:**
- \(\mathbf{0} \in \text{null}(A)\) since \(A\mathbf{0} = \mathbf{0}\).
- If \(\mathbf{x}, \mathbf{y} \in \text{null}(A)\), then \(A(\mathbf{x}+\mathbf{y}) = A\mathbf{x} + A\mathbf{y} = \mathbf{0} + \mathbf{0} = \mathbf{0}\).
- If \(\mathbf{x} \in \text{null}(A)\) and \(c \in \mathbb{F}\), then \(A(c\mathbf{x}) = cA\mathbf{x} = c\mathbf{0} = \mathbf{0}\).

The dimension of the null space is the *nullity* of \(A\). By the rank-nullity theorem: \(\text{rank}(A) + \text{nullity}(A) = n\).Copy