I have $A_1,\ldots,A_k$ matrices of dimension $n \times m$ (they are actually stochastic). Let $\alpha_i \ge 0$ and $\sum_i \alpha_i = 1$. Define $A = \sum_i \alpha_i A_i$.
Under what conditions $\mathrm{range}(A_i) \subseteq \mathrm{range}(A)$ for all $i$?
Is it true, for example, if there is some $j$ such that $\mathrm{range}(A_i) \subseteq \mathrm{range}(A_j)$ for all $i$? (I think not.)
What about if $\mathrm{range}(A_1) \subseteq \mathrm{range}(A_2) ... \subseteq \mathrm{range}(A_k)$? (I think yes.)