Thread: inner product spaces

Can you please, help me with this
Let T: V->W is linear. Prove that
(a) T is injective if and only it T* is surjective.
(b) T is surjective if and only it T* is injective.
Thank you!

Originally Posted by bamby

Can you please, help me with this
Let T: V->W is linear. Prove that
(a) T is injective if and only it T* is surjective.
(b) T is surjective if and only it T* is injective.
Thank you!

Start with the definitions. What is the definition of "injective"? What is the definition of "surjective"? What is the definition of A*?

I am so stupid but I don't see how if T is injective will get T* is surjective. And if T is surjective T* will be injective.
And by A* do you mean the transition matrix?
Thank you!

Start with the definition of T*: (where the angled brackets denote the inner product).

Now ask yourself what does it mean to say that for all vectors y. If the first of those inner products is zero for all y it tells you that Tx must be zero. If the second inner product is zero for all y it tells you that x must be orthogonal (perpendicular) to the range of T*. You should be able to see from this that the kernel of T is {0} if and only if the range of T* is the whole space.