To find a basis for the column space of a matrix one finds the RREF of the matrix. The columns in the RREF are not a basis for the column space, but the same columns in the original matrix are a basis.
I get that the rox space doesn't change under elementary row operations, and I can see (from examples) how the column space does change.
But I can't understand why one can just pick the columns in the original matrix when finding a basis for the column space. Is there a proof of this?