For any positive integer $d$ define the box $B_d$ as all points $(x_1, ..., x_d) \in \mathbb{R}^d$ with $|x_i| < 1.$
Suppose I have some invertible $n \times n$ matrix $M_1$, and for an integer $k<n$ I have the $k \times n$ projection matrix $P$ which projects a vector from $\mathbb{R}^n$ to $\mathbb{R}^k$ by removing the last $n-k$ components.
So I can look at the set $PM_1B_n$, which is transforming and then projecting the box $B_d$. Is it true that $PM_1B_n = M_2B_k$ for some invertible $k \times k$ matrix $M_2$? That is, if I transform and then project the box $B_n$, is that always equal to a transformation of the lower dimensional box $B_k$?
I can't seem to come up with a situation where the projection collapses things to lower than $k$ dimensions, but I am not certain that this should hold either...