Let $a=(a_1,\dots,a_i,\dots,a_n)$ be a probability vector, i.e. $\forall i: a_i\ge0$ and $\sum_i a_i=1$. Suppose $b$ is another $n$-dimensional probability vector.
Is it true that there always exists an $n\times n$ matrix $M$ with non-negative entries and whose columns sum to $1$ such that $b=M^Ta$?
If your $a$ is a row vector, you want to write that as $b = a M^T$.
Then simply take $M$ as the matrix whose columns are all equal to $b^T$.