What is the relationship between the rank of a matrix and its transpose?

[oldmathguy]

Hi, I'm new to the forum but have watched it for some time. I am trying to prove that Rank (A^T) = Rank (A) with A being mxn matrix. I suspect that it has to do with Rank (A) = Row Rank (A) = Column Rank (A) -and- A^T simply being rows / columns transposed but am unsure how to prove. Thanks, John.

 

[morphism]

What's the column rank of A? And what's the row rank of A^t?

What's the relation between these two?

 

[oldmathguy]

They're equal !

Col Rank (A) = Row Rank (A^T) so dim (A) = Col Rank (A) = Row Rank (A^T) = Rank (A^T). Thanks ! John.

 

[mathwonk]

use row reduction. or the general theory of dimension.