The null space of a matrix is a subspace of R^n (or C^n).
Created almost 9 years ago, updated 5 months ago
ID24110
Theorem: The null space of an (m imes n) matrix (A) is a subspace of (\mathbb{F}^n): [ ext{null}(A) = {\mathbf{x} \in \mathbb{F}^n : A\mathbf{x} = \mathbf{0}}]
Proof:
(\mathbf{0} \in ext{null}(A)) since (A\mathbf{0} = \mathbf{0}).
If (\mathbf{x}, \mathbf{y} \in ext{null}(A)), then (A(\mathbf{x}+\mathbf{y}) = A\mathbf{x} + A\mathbf{y} = \mathbf{0} + \mathbf{0} = \mathbf{0}).
If (\mathbf{x} \in ext{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: ( ext{rank}(A) + ext{nullity}(A) = n).
**Theorem:** The *null space* of an \(m imes n\) matrix \(A\) is a subspace of \(\mathbb{F}^n\):
\[ ext{null}(A) = \{\mathbf{x} \in \mathbb{F}^n : A\mathbf{x} = \mathbf{0}\}\]
**Proof:**
- \(\mathbf{0} \in ext{null}(A)\) since \(A\mathbf{0} = \mathbf{0}\).
- If \(\mathbf{x}, \mathbf{y} \in ext{null}(A)\), then \(A(\mathbf{x}+\mathbf{y}) = A\mathbf{x} + A\mathbf{y} = \mathbf{0} + \mathbf{0} = \mathbf{0}\).
- If \(\mathbf{x} \in ext{null}(A)\) 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: \( ext{rank}(A) + ext{nullity}(A) = n\).