IMC 1996 Problem 3

The linear operator $A$ on a finite-dimensional vector space $V$ is called an involution if $A^{2}=I$ , where $I$ is the identity operator. Let $\dim V=n$ . 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 $V$ .