# Determine the image of the hypercube under the linear map

Let $$A$$ bean $$3 N times$$ matrix (where $$N$$ is large) with real non-negative inputs. I would like an algorithm to determine when a vector $$v in Bbb R ^ 3$$ can be written as $$Aw$$ for some vector $$w in Bbb R ^ N$$ at each entry of $$w$$ in the perimeter $$[0,1]$$. In other words, I would like to find the image of the hypercube unit in $$Bbb R ^ N$$ under the map represented by $$A$$. Is there an algorithm that works in polynomial time in $$N$$?

Until now, all I've thought is exponential, basically looping over the $$2 ^ N$$ heights of the hypercube. The problem is similar to that of determining the convex hull of a given set of points, except that the entries in $$w$$ should not be summarized to $$1$$. Obviously take the convex hull of all $$2 ^ N$$ the points would work, but it's again exponential.

My motivation is to determine the range of colors that can be seen under a given light source, depending on the spectral distribution of the power of the light source and the sensitivity curves of the three cells of the cone of the eye.