linear operator on a vector space V such that $T^2 -T +I=0$

Question

let T be a linear operator on a vector space V such that $T^2 -T +I=0$.Then

1. T is oneone but not onto.
2. T is onto but not one one.
3. T is invertible.
4. no such T exists.

could any one give me just hint?

Answer 1

Let $\mathbb{x}$ be any vector in the nullspace. Then $T\mathbb{x} = \mathbb{0}$. Using your equation $T^2 - T + I = 0$, what can you conclude about $\mathbb{x}$?

Alternatively if you know about minimal polynomials: How does your polynomial split?

Answer 2

$$ T^2-T+I=0 \iff T(I-T)=I=(I-T)T, $$ i.e. $T$ is invertible and $T^{-1}=I-T$. In particular $T$ is injective and surjective.

Answer 3

$T(T-1)= -I$ then $detT\cdot det(T-I) = (-1)^n$ which implies $det(T) \neq0$ hence $T$ invertible