Abstract Algebra?

I have two questions, which I think are harder then I think.





1) Let V be a finite-dimensional inner product space, and let T be a linear operator on V. Prove that if T is invertible, then T* is invertible and (T*)^-1 = (T^-1)*.





What I have done is:





TT^(-1) = I = T^(-1)T


Taking the adjoint of all the sides I get:





[T^(-1)]*t* = I* = I = T*[T^(-1)]; I'm using the property (TU)* = T*U*


Which then shows that T* is invertible and its easy to finish it off. But I'm wondering what I did is actually correct, b/c then it would be really easy, and I feel theres more to it.





2) For a linear operator T on an inner product space V, prove that T*T = T_0 implies T = T_0. Is the same result true if we assume that TT* = T_0?





And well this one I'm just having troubles with and really don't know how to do it, I thought of multiplying both sides by T^(-1) but I don't know whether T is invertible.





Help would be greatly appreciated =)

Abstract Algebra?
1. You did the hard work. I'd just make it a little clearer:


[T^(-1)]* T* = [T T^(-1) ]*= I* = I .


T* [T^(-1)]* = [T^(-1) T]*= I* = I .


Thus [T^(-1)]* is both a left and right inverse for T*.





2. I assume that T_0 is supposed to be the linear transformation that maps everything to 0.


If T*T = T_0, then for every x, 0 = x* T* T x = || Tx ||², so Tx=0.


Hence T = T_0.





If TT* = T_0, the same argument proves that T* = T_0. Question for you: does that imply that T = T_0?