I have a matrix $B \in \mathbb{R}^{m \times n}$ such that for all vectors $x$, whose entries are positive and whose sum of entries is equal to $1$ (i.e. $x_i >0$ and $x_1+\dots+x_n =1$), we have $B x \neq 0$. Can I find a vector $y$ such that $y B > 0$, i.e., all entries are greater than or equal to $0$ and at least one is greater than $0$? My goal is to construct an explicit $y$.
At first I thought to define $y := x^T B^T$, since I though all entries in $B^T B$ are sums of squares, which is of course nonsense.