Elementary column operations (True or false) I have been given a statement: Elementary column operations do not necessarily preserve rank.
I know that this is false, but I am not sure if my reasoning is correct.
So, basically, I think the one reason that elementary column operations preserve rank follows from the fact that $\textrm{rank } A = \textrm{rank } A^T. $  It sort of makes sense I guess but I feel like I am making some logical mistakes since I am new to the material.
Any comments will be much appreciated.
I am not asking why the statement is false. I want your opinions on the reason I have in mind. 

The rank of the matrix is, amongst many equivalent characterisations, the number of linearly independent columns that it possesses. It suffices then to consider the manner in which the number of linearly independent vectors, or the size of the basis, of a set of vectors is invariant under the corresponding elementary operations. Under these operations, the spanning set of the vectors clearly does not change and you can recover the original basis. The same argument applies, in particular, to your problem.
