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If $A$ and $B$ are $n \times n$ matrices that have the same column space, then $A$ and $B$ have the same row space.

Can one prove or disprove this? This is my continuation of Same row space is equivalent to same column space?

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2 Answers 2

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False:

Let $A=\begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix}$

and, $B=\begin{pmatrix} 1 & 1\\ 0 & 0 \end{pmatrix}$

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  • $\begingroup$ Any other methods of disproving this without counterexamples? $\endgroup$
    – yolo123
    Aug 25, 2014 at 3:43
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    $\begingroup$ @yolo123: Disprove usually means "give a counterexample that doesn't work". Of course- there might be some theorems that say the conjecture is false. However, in this case- there are examples where the conjecture is true- and hence the best method is to give a counterexample. $\endgroup$
    – voldemort
    Aug 25, 2014 at 4:40
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If two $n\times n$ matrices have the same column space, then there is a sequence of elementary column operations transforming one matrix to the other. But column operations need not preserve the row space of a matrix.

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