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

**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 \(\text{null}(A)\) 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.Copy