A real or complex linear system has zero, one, or infinitely many solutions

For systems over an **infinite field**, the possibilities are no solution, a unique solution, or infinitely many solutions. If $x_1\ne x_2$ both solve $Ax=b$, then $(1-t)x_1+tx_2$ also solves it for every scalar $t$. Distinct $t$ give distinct vectors because $x_2-x_1\ne0$.

For a consistent rank-$r$ system with $n$ unknowns over $\mathbb F_q$, there are instead $q^{n-r}$ solutions. This finite-field exception matters in linear coding.Copy