Zermelo
Junior Member
- Joined
- Jan 7, 2021
- Messages
- 148
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.
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.