I am auditing a Linear Algebra class, and today we were taught about the rank of a matrix. The definition was given from the row point of view:
"The rank of a matrix A is the number of non-zero rows in the reduced row-echelon form of A".
The lecturer then explained that if the matrix $A$ has size $m \times n$, then $rank(A) \leq m$ and $rank(A) \leq n$.
The way I had been taught about rank was that it was the smallest of
- the number of rows bringing new information
- the number of columns bringing new information.
I don't see how that would change if we transposed the matrix, so I said in the lecture:
"then the rank of a matrix is the same of its transpose, right?"
And the lecturer said:
"oh, not so fast! Hang on, I have to think about it".
As the class has about 100 students and the lecturer was just substituting for the "normal" lecturer, he was probably a bit nervous, so he just went on with the lecture.
I have tested "my theory" with one matrix and it works, but even if I tried with 100 matrices and it worked, I wouldn't have proven that it always works because there might be a case where it doesn't.
So my question is first whether I am right, that is, whether the rank of a matrix is the same as the rank of its transpose, and second, if that is true, how can I prove it?