If $\mathbb{R}^n=\operatorname{span}\{X_1,X_2,...,X_k\}$ and $A$ is an $m\times n$ matrix, does $AX_i$ not equal $0$ for some $i$?
I think the answer might be no. But when I tried to prove it, I notice the way I did is not so persuasive. Can anybody help?