What ideas have you had so far?
I'm stuck on this h/w problem...any help would be appreciated!!! thank u
Let V be a vector space of dimension n, and T: V --> V an invertible operator. Prove that if U is a T-invarient subspace of V, then U is also invariant under T^-1. Justify each step.
Well I don't know too much about T-invarient subspaces, but I figured it means eigenvalues. So here is my prrof so far-
First suppose that U is an eigenvalue of T. Thus there exists a nonzero vector v exists in V such that: Tv=Uv
Appyling T^-1 to both sides of the equation above and we get v=UT^-1v, which is equivantly to the equation T^-1v=(1/U)v. Thus (1/U) is an eigenvalue of T^-1. Therefore proving that if U is a T-invarient subspace of V, then U is also invariant under T^-1.
What do u think??
Thank you.
No, I don't think you understand what's going on. A -invariant subspace is a subspace of that remains the same under action from . That is, . This relation says that it is the case that
Can you write down, now, a mathematical statement that represents what you need to prove?