Suppose a have a polytope $P$ defined by $Apmb{x} leq pmb{b}$. Suppose $pmb{x} in mathcal{X} = { pmb{x} in mathbb{R}^d | l_i leq x_i leq r_i, i = 1,ldots, d}$ i.e a hyperrectange in $mathbb{R}^d$. How would $A$, $pmb{b}$ change if I were to apply a linear rescaling to $pmb{x}$ 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 $Apmb{x} leq pmb{b}$ 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 $pmb{x}^{*} in (0,1)^d$ then simply rescale: $pmb{x} = text{diag}(pmb{r}-pmb{l})pmb{x}^{*} + pmb{l}$ 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?