Linear Algebra Done Right · Problem 3E.8

The key insight is that $Tx = c$ describes an affine subspace rather than a linear subspace—it's shifted away from the origin unless $c = 0$. If any particular solution $x_0$ exists, then every other solution differs from it by an element of $\operatorname{null} T$ (where $T$ acts trivially), so the solution set is $x_0 + \operatorname{null} T$. For linear systems, the same principle applies: the coefficient matrix $A$ plays the role of $T$, and the solution set is either empty or a translate of $\operatorname{null} A$.

The key insight is that $Tx = c$ describes an affine subspace rather than a linear subspace—it's shifted away from the origin unless $c = 0$. If any particular solution $x_0$ exists, then every other solution differs from it by an element of $\operatorname{null} T$ (where $T$ acts trivially), so the solution set is $x_0 + \operatorname{null} T$. For linear systems, the same principle applies: the coefficient matrix $A$ plays the role of $T$, and the solution set is either empty or a translate of $\operatorname{null} A$.