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.