inner product spaces

• Oct 18th 2008, 10:24 PM
bamby
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!
• Oct 19th 2008, 05:19 AM
HallsofIvy
Quote:

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*?
• Oct 25th 2008, 07:38 PM
bamby
inner product spaces
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!
• Oct 26th 2008, 03:00 AM
Opalg
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.