That is the correct definition.
Now why do some columns not have pivots? Think hard about this. Look at concrete examples and see if the columns that do not have a pivot in them are in fact linear combinations of the columns that do have pivots in them. After looking at a few examples think about why this happens.
Please post back showing us the work you did for a couple of matrices like I asked for above. Include your reason why you think this happens. If you do not see why, then we'll give you another hint.