Can linear transformation of variables cause unboundedness of a linear programming?

Question

Let $x$ be a $n$ $\times$ $1$ variable. Then I have a linear programming which is assumed to be bounded and has a solution:

$$\max c^Tx$$ $$s.t. Ax \leq b $$

Now I apply a linear transformation on $x$, i.e., $x$ now becomes $Tx$, where $T$ is the transformation matrix. Then the linear programming becomes: $$\max Tc^Tx$$ $$s.t. ATx \leq b$$

Here I am curious as to whether there would exist a $T$ that causes this new LP to be unbounded.

Thanks.

Answer 1

No. By boundedness, you have $x\leq U$ for all $x$ such that $Ax\leq b$ for some bound $U$. So if $ATx\leq b$, then $y=Tx$ satisfies $Ay\leq b$, and thus $Tx=y\leq U$.