One way to prove that is to show that , which is what I did. If are linearly dependent then must be linearly dependent. Thus if is linearly independent we must necessarily have that are also linearly independent.
The proof is given. Which part don't you understand?
if and were dependent, ,therefore and will be in with a dimension less than 2,hence they can't be independent.
is that what you're trying to say(Thinking)
P.S: i like your proof i just can't see through it.(Headbang)