I'm not sure if "reverse matrix dot product" is the proper term I'm looking for so let me explain to the best of my ability. We all know what the dot product is:
$$ a \cdot b = \sum_{i=1}^{k} a_i b_i $$
However, let's take the example from Wikipedia:
$$ \begin{bmatrix} 1 & 3 & -5 \end{bmatrix} \begin{bmatrix} 4 \\ -2 \\ -1 \end{bmatrix} = 3 $$
Now, let's make this simpler:
$$ A \cdot B = C $$
Suppose we know A and C. Is there a way to find B? Or more specifically, the magnitude of B? Algebraically, this would look like:
$$ x_1 + 3x_2 - 5x_3 = 3 $$
And we are assuming x have to be integers. I do realize there could be an infinite amount of solutions. So now let's restrict the x can only be 0 or 1 and C can only be between 0 and the sum of the values of A inclusive. I believe now there can only be one solution if all the values in A are unique and are not a linear combination of one another. Any advice would greatly be appreciated.