I am working with polytopes that are defined by half-planes in [itex]\Re^N[/itex]. So they are defined by a number of inequalities (half plane representation), but can also be represented by the intersection points of these half planes (vertex representation). Computing the vertices is expensive, so I prefer to do that only when necessary. I need to apply a transformation from this space into one of the same dimension. The transformation is just a linear one via matrix multiplication. The matrix happens to be formed as a product of three non-square matrices. I could provide more detail about this, but in the end, the final matrix used for transportation (the product) is invertible. The polytope I start with is convex. Is there a straightforward argument that shows this transformation will preserve that convexity? When I imagine a planar example, it seems like the stretching necessary to create concave dimples or change the angle of a corner point would have to be nonlinear. I also can't imagine a composition of shift, scale and rotation operations that would do this. I would like to find an argument from first principles or a check to perform on the matrices. Any ideas or solutions would be appreciated.