Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof at http://en.wikipedia.org/wiki/Rank_of_a_linear_transformation and I understand the proof, but I don't "get it". Can someone help me out with this ? 

I find it hard to wrap my head around the idea of how the column space and the row space is related at a fundamental level.
You can apply elementary row operations and elementary column operations to bring a matrix $A$ to a matrix that is in both row reduced echelon form and column reduced echelon form. In other words, there exist invertible matrices $P$ and $Q$ (which are products of elementary matrices) such that
$$PAQ=E:=\begin{pmatrix}I_k\\&amp;0_{(n-k)\times(n-k)}\end{pmatrix}.$$
As $P$ and $Q$ are invertible, the maximum number of linearly independent rows in $A$ is equal to the maximum number of linearly independent rows in $E$. That is, the row rank of $A$ is equal to the row rank of $E$. Similarly for the column ranks. Now it is evident that the row rank and column rank of $E$ are identical (to $k$). Hence the same holds for $A$.
Maybe this helps a bit with the intuition: When you transpose a matrix you don't change the dimension of the image.  But when you transpose a matrix, the column rank becomes the row rank and vice versa. As the dimension of the image is the column rank those are equal.