How to show that T is invertible in this case.

Question

Let $T$ be a linear transformation from a vector space $V$ over reals into $V$ such that $T-T^2=I.$ Show that $T$ is invertible.

I am not able to understand how is $T$ being treated like a matrix when it is infact a linear transformation.

Is it the Matrix equivalent of $T$ being talked about?

Answer 1

Since $T$ is a linear transformation, $T^2$ means $T\circ T$. And$$T-T^2=\operatorname{Id}\iff T\circ(\operatorname{Id}-T)=\operatorname{Id}.$$Therefore, $T$ is invertible and $T^{-1}=\operatorname{Id}-T$.