transformation question

• Aug 14th 2011, 08:48 AM
ibnashraf
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?
• Aug 14th 2011, 10:34 AM
ModusPonens
Re: transformation question
$T^2-T=-I$. Now see if the kernel has any vectors different from zero.
• Aug 14th 2011, 10:46 AM
HallsofIvy
Re: transformation question
Slight variation: $T- T^2= T(I- T)= (I- T)T= I$
• Aug 14th 2011, 11:04 AM
ModusPonens
Re: transformation question
Can you commute T and I-T?
• Aug 14th 2011, 01:40 PM
HallsofIvy
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.
• Aug 14th 2011, 05:45 PM
Drexel28
Re: transformation question
Quote:

Originally Posted by HallsofIvy
Slight variation: $T- T^2= T(I- T)= (I- T)T= I$

Quote:

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.