I'm having trouble understanding something from my course notes about the rank of an mxn matrix:

My prof stated that the rank of an mxn matrix A is equal to the rank of $\displaystyle L_A$ which is in turn equal to $\displaystyle dim(R(L_A))$.

He also said that for an mxn matrix A, rank(A) = r where $\displaystyle r\le{min(m,n)}$.

What I don't understand is how both statements can both be true (probably because I have an incorrect understanding of the first one since I never fully understood $\displaystyle L_A$). I thought that $\displaystyle L_A$ was simply an mxn matrix which converts a nx1 matrix to an mx1 matrix... Is this not true? I doubt it is because if it's true then according to the first statement, the rank would always be equal to 1!

Can someone please set me straight?