How does a linear rescaling alter a polytope equation?

Suppose a have a polytope $$P$$ defined by $$Ax leq b$$. Suppose $$x in mathcal{X} = { x in mathbb{R}^d | l_i leq x_i leq r_i, i = 1,ldots, d}$$ i.e a hyperrectange in $$mathbb{R}^d$$. How would $$A$$, $$b$$ change if I were to apply a linear rescaling to $$x$$ so that $$x_i^{*} = theta_0 x_i + theta_1$$?

In particular, I have a sampling scheme in mind for the design and analysis of a computer experiment – I want to compute a maximin design with boundaries defined by a polytope (see section 5.5 of this book). My parameter space is defined by a polytope $$Ax leq b$$ but it is much more conveniet to construct the design with all the $$x_i$$ scaled to the unit interval (solves the problem of inputs ‘dominating’ each other. Hence, I want to construct my design points on $$x^{*} in (0,1)^d$$ then simply rescale: $$x = text{diag}(r-l)x^{*} + l$$ to generate inputs to a computer model .

In essence, if I linearly rescale my hyperrectangle to be a unit hypercube, how does this change the inequality which defines my polytope?