Precursor to problem: Preparing for an exam in General Equilibrium Theory. One of the important theorems is the No Arbitrage Theorem which states that for an economy with $S$ states of the world and $J$ assets that the asset prices, $q \in \mathbb{R}^{J}$ and a matrix that dictates asset returns (dependent on state), $A \in \mathbb{R}^{S} \times \mathbb{R}^{J}$ that:
No Arbitrage Theorem: For $W := \begin{bmatrix} -q \\ A \end{bmatrix}$. The $\text{Span}(W) \cap \mathbb{R}^{S+1}_+ = \{0 \}$ if and only if $\exists \hat{\pi} \in \mathbb{R}^{S+1}_{++}$ such that $\hat{\pi} W = 0$.
Question: It looks like if the span of a matrix intersected by the non negative numbers is 0 then there is a postive vector in the null space. Is that true and is there an intuitive reason for that? If not, are there any specific conditions on a matrix, $W$ such that we guarantee there is a strictly positive vector (not necessarily unique) in the null space? This is quite possibly a trivial question, but I couldn't find any specific conditions when I looked.
This is more of a comment. The "linear algebraic" perspective from which you're asking the question doesn't pay here when it comes to the first fundamental theorem of asset pricing. (I am abusing language here, of course. Everything is linear algebraic.)
The proper points of view are more geometric. One has to do with the geometry of the Banach space of the appropriate class of stochastic processes. The second one has to do with the geometry of state-dependent payoffs and the portfolio.
The Banach space here is the space of adapted stochastic processes. You're in the simplest two-period setting. There is no arbitrage iff the subspace of all possible divident processes given by $(-q, A)$ intersects the cone of positive adapted processes only at zero. This is your $\text{Span}(W) \cap \mathbb{R}^{S+1}_+ = \{0 \}$ condition.
In the finite dimensional case, there is no arbitrage iff $q$ lies in the cone generated by the state-dependent payoff's (your second condition). Otherwise, any separating hyperplane gives you an arbitrage portfolio. Take the simplest possible case: two assets, two periods, one state. Say $q = (P_1, P_2), W = (P_1', P_2')$. $q$ does not lie in the cone iff $\frac{P_1'}{P_1} \neq \frac{P_2'}{P_2}.$ Then clearly there is arbitrage. Equivalently, $q$ is the discounted expected payoff with respect to an equivalent martingale measure, which is a statement that generalizes to the infinite dimensional setting.