Asked • 05/22/19

Is the rank of a matrix the same of its transpose? If yes, how can I prove it?

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 $m * n$, 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? Thanks :)

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