# Linear transformation over a set of linear inequalities

Let $$A\in\mathbb{R}^{m \times n}$$, $$b\in\mathbb{R}^m$$ such that $$Ax\leq b$$. Now let $$\hat{x} = Tx$$. What can be said about $$A\hat{x}$$? Concretely, is there an analytical way (without using $$T^{-1}$$) for computing a bound $$\hat{b}$$, such that $$ATx\leq\hat{b}$$?