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The solutions to a homogeneous system of linear equations is a vector space.

Created over 8 years ago, updated 10 days ago

Theorem: The set of solutions to a homogeneous system of linear equations (A\mathbf{x} = \mathbf{0}) is a vector space (a subspace of (\mathbb{F}^n), where (n) is the number of variables).

Proof: Let (S = {\mathbf{x} : A\mathbf{x} = \mathbf{0}}).

  • If (\mathbf{x}, \mathbf{y} \in S), then (A(\mathbf{x} + \mathbf{y}) = A\mathbf{x} + A\mathbf{y} = \mathbf{0} + \mathbf{0} = \mathbf{0}), so (\mathbf{x} + \mathbf{y} \in S).
  • If (\mathbf{x} \in S) and (c \in \mathbb{F}), then (A(c\mathbf{x}) = cA\mathbf{x} = c\mathbf{0} = \mathbf{0}), so (c\mathbf{x} \in S).

This space is the null space of (A), denoted ( ext{null}(A)) or (\ker(A)).

Example: The solutions to (x + y + z = 0) form a plane through the origin in (\mathbb{R}^3), which is a 2-dimensional subspace.