This is a please check my proof question. It's not homework.
Given a matrix $A_{n\times m}$ over some field, and its reduced row echelon form $R_{n\times m}$, show that the columns of $A$ corresponding to the columns in $R$ with pivots form a basis for the column space of $A$, $C(A)$.
And of course if anyone has a more elegant proof I'd be happy to see it.