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 $V$.
### Part (i)
1. **Definition and Setup**:
   - Let \( A \) be an involution on \( V \), meaning \( A^2 = I \), where \( I \) is the identity operator.
   - We need to show that there exists a basis of \( V \) consisting of eigenvectors of \( A \).

2. **Eigenvalues of \( A \)**:
   - Since \( A^2 = I \), the eigenvalues \( \lambda \) of \( A \) must satisfy \( \lambda^2 = 1 \).
   - Therefore, the possible eigenvalues of \( A \) are \( \lambda = 1 \) and \( \lambda = -1 \).

3. **Decomposition of \( V \)**:
   - Let \( V_1 \) be the eigenspace corresponding to the eigenvalue \( 1 \), i.e., \( V_1 = \{ v \in V \mid Av = v \} \).
   - Let \( V_{-1} \) be the eigenspace corresponding to the eigenvalue \( -1 \), i.e., \( V_{-1} = \{ w \in V \mid Aw = -w \} \).

4. **Direct Sum of Eigenspaces**:
   - Any vector \( x \in V \) can be written as \( x = \frac{x + Ax}{2} + \frac{x - Ax}{2} \).
   - Define \( v = \frac{x + Ax}{2} \) and \( w = \frac{x - Ax}{2} \).
   - Notice that \( Av = A\left(\frac{x + Ax}{2}\right) = \frac{Ax + A^2x}{2} = \frac{Ax + x}{2} = v \), so \( v \in V_1 \).
   - Similarly, \( Aw = A\left(\frac{x - Ax}{2}\right) = \frac{Ax - A^2x}{2} = \frac{Ax - x}{2} = -w \), so \( w \in V_{-1} \).
   - Therefore, \( x = v + w \) with \( v \in V_1 \) and \( w \in V_{-1} \).

5. **Basis of Eigenvectors**:
   - Since \( V_1 \) and \( V_{-1} \) are eigenspaces corresponding to distinct eigenvalues, they are linearly independent.
   - The direct sum \( V = V_1 \oplus V_{-1} \) spans \( V \).
   - Hence, there exists a basis of \( V \) consisting of eigenvectors of \( A \).

### Part (ii)
1. **Commuting Involutions**:
   - We need to find the maximal number of distinct pairwise commuting involutions on \( V \).

2. **Simultaneous Diagonalization**:
   - Recall that any set of pairwise commuting, diagonalizable matrices is simultaneously diagonalizable.
   - Since involutions are diagonalizable with eigenvalues \( \pm 1 \), we can assume that all the matrices are diagonal in some common basis.

3. **Diagonal Entries**:
   - In the common basis, each involution \( A_i \) can be represented as a diagonal matrix with entries \( \pm 1 \).
   - Each diagonal entry can independently be \( 1 \) or \( -1 \).

4. **Number of Distinct Involutions**:
   - For an \( n \)-dimensional vector space \( V \), there are \( n \) diagonal entries.
   - Each entry can be independently \( 1 \) or \( -1 \), leading to \( 2^n \) possible distinct diagonal matrices.
   - Therefore, the maximal number of distinct pairwise commuting involutions on \( V \) is \( 2^n \).

The final answer is \( \boxed{ 2^n } \).