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!

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- Oct 18th 2008, 10:24 PMbambyinner 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 AMHallsofIvy
- Oct 25th 2008, 07:38 PMbambyinner 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 AMOpalg
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.