I believe this is true because when we have unconstrained independent variables, we must pivot each one and file it in order to convert the tableau to canonical form.
Now, if we have more columns than rows in a tableau, and we must pivot on each column, then it suffices to say that the resulting solution would be unbounded.
I am not sure if this proof is actually correct.
This is False, consider the following feasibility model:
$$\max z = 0$$ $$\text{Subject to:}\qquad\qquad\qquad\qquad\qquad\qquad$$ $$x+y+w\le10$$ $$x,y,w\in\mathbb{R}^3$$
Here, we have three free variables and one constraint, and the Simplex method is capable of solving this via free-variable substitution:
$$\max z = 0$$ $$\text{Subject to:}\qquad\qquad\qquad\qquad\qquad\qquad$$ $$x' - x'' + y' - y'' + w' - w'' \le 10$$ $$x',x'',y',y'',w',w''\ge0$$
where $x = x' - x''$, $y = y' - y''$, and $w = w' - w''$. The goal of the Simplex method is not to give us the most "optimal" solution, as the term optimal means almost nothing for the model since the objective function is a constant, but the Simplex method will terminate once it finds a value that satisfies the single constraint (thus, for this model there are infinite solutions, for example this solver reports $x=10$, $y=0$, and $w=0$, but $x=0$, $y=5$, and $w=5$ works and so on).
Tangential, but this whole process and logic is called feasibility modeling, as the objective function, $z$, is unnecessary as we're only looking for solutions that satisfy the constraints. Famous constraint satisfaction problems are:
