Here in my notes I have an example of getting the columnspace of a matrix A. The answer is all linear combinations of (1,0,0,1) and (-2,1,0,0).
But I know that you can get the columnspace by reducing a matrix to echelon form and then looking at the pivot columns - the columns in the original matrix that correspond to the pivot columns will be a basis for the columnspace. If I do that in the case of the example below I get all linear combinations of (1,0,0,1) and (0,1,0,2). These vectors seem to span the same space as the vectors above - (1,0,0,1) and (-2,1,0,0). So are there two ways of getting the columnspace for a matrix?