1. transformation question

Question:

If T is a linear operator on a vector space V(F) such that $T^2-T+I=Z$ (zero map), then show that T is invertible.

i know that to show invertible i have to show 1-1 and onto, but i have no idea how to begin showing this ... any help?

2. Re: transformation question

$T^2-T=-I$. Now see if the kernel has any vectors different from zero.

3. Re: transformation question

Slight variation: $T- T^2= T(I- T)= (I- T)T= I$

4. Re: transformation question

Can you commute T and I-T?

5. Re: transformation question

Yes, of course: $T- T^2= I*T- T*T= T*I- T*T$. Any linear operator from a vector space to itself commutes with itself and the identity.

6. Re: transformation question

Originally Posted by HallsofIvy
Slight variation: $T- T^2= T(I- T)= (I- T)T= I$
Originally Posted by ModusPonens
Can you commute T and I-T?
Moreover, judging from the level of the question one would guess that one can assume that the vector space is finite dimensional from where $T(\mathbf{1}-T)=\mathbf{1}$ implies $T$ is surjective from where (by finite dimensionality) bijectivity follows.