Injective linear operator question

[Zermelo]

Hello there, my Linear Algebra textbook says an interesting thing: "A linear operator T: V ->W is injective if and only if there doesn't exist any element x from V, [MATH]x \neq 0v[/MATH] such that T(x) = 0w".
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 [MATH]x \neq 0v[/MATH] 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 [MATH]x \neq y[/MATH] 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.

 

[Steven G]

Fact: T: V ->W is injective if and only if there doesn't exist any element x from V, x≠0v such that T(x) = 0w".

Proof: (<=)Suppose T(a) = T(b). Then T(a) - T(b) = 0w. So T(a-b) = 0w. Since only T(0v) = 0w, it follows that a-b = 0v or a=b. But if a=b, then T is 1-1.


You do realize that there doesn't exist any element x from V, x≠0v such that T(x) = 0w means that only T(0)=0w?

 

[Zermelo]

Fact: T: V ->W is injective if and only if there doesn't exist any element x from V, x≠0v such that T(x) = 0w".

Proof: (<=)Suppose T(a) = T(b). Then T(a) - T(b) = 0w. So T(a-b) = 0w. Since only T(0v) = 0w, it follows that a-b = 0v or a=b. But if a=b, then T is 1-1.


You do realize that there doesn't exist any element x from V, x≠0v such that T(x) = 0w means that only T(0)=0w?

Nice! Now I do realize. Thanks a lot!