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

Created over 8 years ago, updated 10 days ago

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).