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 definition of T*: $\displaystyle \langle Tx,y\rangle = \langle x,T^*y\rangle$ (where the angled brackets denote the inner product).
Now ask yourself what does it mean to say that $\displaystyle \langle Tx,y\rangle = \langle x,T^*y\rangle = 0$ 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.