While the "if", aka "If T is injective, there doesnt exist..." makes sense and is pretty trivial, but I don't see any reason for the "only if".

Let's suppose that an element \(\displaystyle x \neq 0v\) such that T(x) = 0w doesn't exist. Why does this imply that the operator is injective? I really don't see a problem with having some two arbitrary elements x and y from V such that \(\displaystyle x \neq y\) and T(x) = T(y) under these circumstances. Am I missing something or did my professor make a mistake?

NOTE: 0v and 0w are the neutral elements of vector spaces V and W, respectfully.