I understand how to determine WHETHER a set of vectors span a space, but I've been given two vectors (1,2,3,4) and (17,17,17,17) and asked to find two more vectors such that the set of all these vectors span $ \mathbb{R}^4 $.
At the moment I'm just going about with kind of a trial and error approach…is there some better way to work this out?
Hints, or suggestions would be greatly appreciated as I want to figure this out without a full solution.
Best Answer
Try to complete your basis using the vectors of the standard basis: $(1,0,0,0), (0,1,0,0),$ etc. Pick one of them and check if the three vectors you have now are independent. If yes, great, keep the vector you chose and repeat the procedure with the three remaining vectors of the standard basis. If not, discard the vector you chose and try again with another one. Repeat until done.