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For some vectors $v_1, v_2$ and a matrix $A$, I currently have that $v_1 A = v_2$. I want a representation that has $A$ on the left of $v_1$, but $A$ can be modified. I'm looking for some $A'$ such that $A' v_1 = v_2$.

What modifications do I have to make to $A$ to get such a representation?

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Normally $vA$ is meaningful only if $v$ is a row vector, and $Av$ is meaningful only if $v$ is a column vector. So (for vectors with more than one component) there are no cases where your question is meaningful. – Marc van Leeuwen Apr 17 '12 at 16:09
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I assume v1, v2 are 1 x n vectors and A is n x n. Then you would simply write $A^Tv_1^T = v_2^T$. Am I missing something here?

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