Theorem: Finite alternative

For a matrix $N\in K^{s\times L}$, a vector $u\in K^s$, and a field $K$, exactly one of the following holds:

or

The second case is equivalent to $\mathrm{rank}[N\mid u]=\mathrm{rank}N+1$.

Proof. A fit makes every left annihilator of $N$ annihilate $u$. If $u\notin \mathrm{im}N$, append $u$ to a basis of $\mathrm{im}N$, extend to a basis of $K^s$, and define a linear functional equal to one on $u$ and zero on the other basis vectors. Appending a single column increases rank exactly when that column lies outside the original image.