Could non pivot columns form the basis for the column space of a matrix?

Since the pivot columns of a matrix are always linearly independent and span the columns of the matrix, they usually make up the basis of the matrix. However, this had me wondering about the non pivot columns of the matrix. According to column correspondence theory, the non pivot columns are linear combinations of the pivot columns. Does this mean that the non pivot columns could also be a generating set of the columns like how the pivot columns are a generating set of the columns? Then, if the non pivot columns were proven to be linearly independent by finding the reduced row echelon form or by some other means and the number of non pivot columns was equal to the number of pivot columns (therefore, the dimension wouldn't be a problem), could the non pivot columns make up a basis for the column space of a matrix?

• @Bye_World: I suggest that you answer in the answer box and not in the comments. Nov 24, 2015 at 11:31
• @Bye_World Why did you delete your comment? I thought it was a good answer :( Nov 24, 2015 at 15:10