I had the following statement to prove:
Let be a linear transformation, and a set of vectors.
If are linear independent, and is one-to-one, then are also linear independent.
that's how I proved it:
are linear independent, so if:
now, since is a linear transformation, it holds that , therefore:
and since we know that , then are linear independent.
my question here is: why should I care that is one-to-one?
any help would be greatly appreciated.