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*: (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.