Just looking at it quickly here, I have some studying to do for a final in a few hours, I would take a look at that 4th vector you have there. I noticed it has a lot of 0s in it. You need only show that there is a 5x5 minor with nonzero determinant.

I would look at the minor determined by the 4 rows with 0s in that position in column 4 and then try to carefully pick that last 5th row to take the determinant of that 5 x 5 resulting matrix. It should be pretty easy by cofactor expansion with all those 4 out of 5 entries being 0 in that column that you picked.

No guarantee that this is going to be the correct minor though, but as it appears it was an assignment, my guess is there should be an easy way to do it other than computing determinants of all 5x5 minors of that thing.

Hope it was of some assistance.