📐 Homogeneous System Solutions

The solution set of a homogeneous system $Ax = \\mathbf{0}$ is a subspace of $F^n$, called the null space of $A$.

Proof: Let $N = \\{x \\in F^n : Ax = \\mathbf{0}\\}$.

- $A \\cdot \\mathbf{0} = \\mathbf{0}$, so $\\mathbf{0} \\in N$.

- If $x, y \\in N$, then $A(x + y) = Ax + Ay = \\mathbf{0} + \\mathbf{0} = \\mathbf{0}$.

- If $x \\in N$ and $c \\in F$, then $A(cx) = c(Ax) = c \\cdot \\mathbf{0} = \\mathbf{0}$.

From: Advanced Linear Algebra

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