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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?

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Are there any conditions on $A$? –  Dylan Moreland Jan 27 '12 at 3:47
NO, just an m*n matrix. –  Shannon Jan 27 '12 at 3:51
Do you know of any $m \times n$ matrices that send everything to zero? –  Dylan Moreland Jan 27 '12 at 3:53
Do you mean 0 matrix? This question would be meaningless if A is 0 matrix. –  Shannon Jan 27 '12 at 4:43
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1 Answer

Note that if $AX_i=0$ for all $i$ then $A(a_1X_1+...+a_kX_k)=0$. Thus, as $span\{X_1,...,X_k\}=\mathbb R^n$, it follows that $AX=0$ for all $X\in \mathbb R^n$. Therefore $AX_i=0$ for all $i$ if and only if $A$ is the zero-matrix.

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The question is that is it possible for SOME i AX_i could not equal 0? Assume we can pick arbitrary nonzero X_i from R^n,but What I am concerned is that the product of two matrices also could be zero. –  Shannon Jan 27 '12 at 4:49
If $A$ is the zero-matrix then clearly all $AX_i$ are zero but for any other matrix it is true. –  azarel Jan 27 '12 at 4:53
Talked to the prof today that he made some revision and stated A should be nonzero matrix. Thank you all. –  Shannon Jan 27 '12 at 22:14
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