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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.)

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Let $A_1=B$, $\alpha_1=1/2$, and $A_i=-B$, $\alpha_i=1/(2(n-1))$ for every $i\geqslant2$, for a given nonzero $B$.

Then $A=0$ hence $\mathrm{range}(A_i)\subseteq\mathrm{range}(A)$ cannot hold for any $i$, but $\mathrm{range}(A_i)$ does not depend on $i$, in particular $\mathrm{range}(A_1)\subseteq\mathrm{range}(A_2)\subseteq\cdots\subseteq\mathrm{range}(A_n)$.