Let \(V\) be a real vector space and \(E\) an idempotent linear operator on \(V\), i.e., a projection. Prove that \((I + E)\) is invertible. Find \((I + E)^{-1}\).
Solution
Because \(E^2=E\), \[ \left(I-\frac12E\right)(I+E) =I+E-\frac12E-\frac12E^2 =I. \] The same computation in the opposite order gives \[ (I+E)\left(I-\frac12E\right)=I. \] Since the scalar field is \(\mathbb R\), the scalar \(2\) is invertible. Therefore \[ \boxed{(I+E)^{-1}=I-\frac12E.} \] Thus \(I+E\) is invertible [@HK71].