Let A and B be matrices having more rows than columns. It's obvious that if [A,B] has full column rank, then A and B have full column rank. But what if [A,B]^T has full column rank. Then it doesn't imply that A or B have full column rank, right? (Also the converse direction doesn't work, right?)

Let A and B be matrices having more rows than columns. It's obvious that if [A,B] has full column rank, then A and B have full column rank.
But what if $\left[A,B{\right]}^{T}$ has full column rank. Then it doesn't imply that A or B have full column rank, right?
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Carson Mueller
Let $A=\left[\begin{array}{cc}1& 0\\ 0& 0\\ 0& 0\end{array}\right]$ and $B=\left[\begin{array}{cc}0& 0\\ 1& 0\\ 0& 1\end{array}\right]$. Then $\left[A,B{\right]}^{\mathrm{\top }}$ has full column rank but A does not.