In the DFM \(\boldsymbol{z}_{t}=\boldsymbol{P} \boldsymbol{f}_{t}+\boldsymbol{a}_{t}\) with \(\boldsymbol{P}=\frac{1}{\sqrt{k}} \mathbf{1}\) is a \(k \times 1\) vector, \(\mathbf{1}=(1, \ldots, 1)^{\prime}\) and \(\boldsymbol{a}_{t}\) is white noise with \(\boldsymbol{\Sigma}_{a}=\sigma^{2} \boldsymbol{I}\), (1) explain the form of the linear combinations of the series \(\boldsymbol{c}_{j}^{\prime} z_{t}\) for \(j=1, \ldots, k-1\) that are white noise; (2) find the variance of these linear combinations.