Prove that an underdetermined system of cannot have a unique solution(Is this proof correct?) I know I misspelled underdetermine but is this proof correct? How can I improve it either way? !enter image description here Side Remark: Anyone who is down-voting please can you understand I new to this site and somewhat anxious. I wouldn't mind the down vote but they are threating to close my account. I don't see why a down vote is waranted unless I was deliberately wasting time or it must be that this community doesn't encourage learning.

Your proof is mostly correct, but you're missing an important piece.

You've accurately stated that, if the system is consistent, then (because of the presence of basic variables) there are infinitely many solutions. You should additionally say that if the system is _inconsistent_ , then there by definition is _no_ solution, and so we still can't have a unique solution.