IMC 1996 · Problem 3

The linear operator $A$ on the vector space $V$ is called an involution if $A^2 = E$ where $E$ is the identity operator on $V$. Let $\dim V = n < \infty$.

(i) Prove that for every involution $A$ on $V$ there exists a basis of $V$ consisting of eigenvectors of $A$.

(ii) Find the maximal number of distinct pairwise commuting involutions on $V$.