Why is the product of a matrix variable and vector variable affine? You can constrain the product to equal an affine vector and constrain column sums to 1, but that won’t make the products be affine. Perhaps you have some particular meaning in mind which enables you to “prove that this is is affine by using linearity, the positivity of every entry and the column sum condition.”,; but if so, I have no idea what you mean,
if you have some underlying convex optimization (or feasibility) problem, you need to make clear exactly what that mathematical problem is. I have no idea what problem you are trying to model or solve.