Let $T : \Bbb R^n \rightarrow \Bbb R^n$ Linear transformation.

Prove that there is a real number $\alpha$ that the transformation $\alpha I-T$ is isomorphism.

isomorphism is only if $\ker T={0}$ **or** $\text{Im } T=\Bbb R^n$ because the transformation is from the same dimension.

If $0$ is an eigenvalue then $T$ isn't isomorphism, so I just need to choose $\alpha \neq 0$?