Same row space is equivalent to same column space? If $A$ and $B$ are $n \times n$ matrices that have the same row space, then $A$ and $B$ have the same column space.
This is false of course. I could just come up with examples though. Can one prove this?
 A: Theory tells us that if the row space of $A$ equals the row space of $B$, then the ranks of the column spaces of $A$ and $B$ are equal.  Let this guide us to a minimal counterexample.
Clearly if the dimensions of the column spaces were zero, then indeed the columns spaces would be trivial (and equal).  So the minimal counterexample involves dimension (at least) one.
What do rank one matrices with the same row space as $A$ look like?  Do any of them have different columns spaces?  (Hint: yes.)
If you know matrix multiplication, here's an easy way to build a pair of matrices with the same one-dimensional row space but different column spaces.  Pick a nonzero row vector $v$ of length $n$.  Pick two nonzero column vectors $u^T$ and $w^T$, also of length $n$, which are not scalar multiples of one another (so the spaces spanned by $u^T$ and $w^T$ are not equal).
Let $A = u^T v$ and $B = w^T v$.  Then the row spaces of $A$ and $B$ are both the one-dimensional space spanned by $\{v\}$, but the column spaces are different.  The column space of $A$ is the space spanned by $\{u^T\}$, while the column space of $B$ is the space spanned by $\{w^T\}$, and we just arranged that these would be distinct.
Note that all the rows of $A$ and all the rows of $B$ are scalar multiples of $v$, and in each case there is at least one nonzero row.  This proves the claim about the row spaces of $A$ and $B$ being the same.  Similarly the column space of $A$ is the span of $\{u^T\}$ and the column space of $B$ is the span of $\{w^T\}$.
Any doubt about reversing the roles of row and column spaces should be dispelled by taking the transposes of $A$ and $B$, or $A^T$ and $B^T$ as we usually denote them.  These last two $n\times n$ matrices will have equal column spaces but different row spaces.
