I was solving the following question from my textbook but couldn't conclude an answer.
Is it possible for a nonhomogeneous system of seven equations in six unknowns to have a unique solution for every right-hand side?
$Ax=b\\$, where A is a 7 x 6 matrix. Since n = 6, $Rank(A)\leq6$
Because there are 7 equations, $m=7$, thus $Col(A)$ is a subspace of $\mathbb{R^7}$. Since $Rank(A)$ is at most 6, $Col(A)$ is at most a 6-dimensional subspace of $\mathbb{R^7}$.
At this point, I'm not sure what to conclude from this reasoning.
That's is: since $\operatorname{col}A$ has dimension $\le6$, it isn't the whole $\Bbb R^7$, and therefore there is a vector $b\in\Bbb R^7\setminus\operatorname{col}A$. Any such vector satisfies $[\forall x\in\Bbb R^6, Ax\ne b]$.