= Solution
For a <sum of mutually orthogonal projectors>, <orthonormality> gives $q_i^Tq_j=0$ for $i\ne j$ and one for $i=j$. Therefore $P_iP_j=q_i(q_i^Tq_j)q_j^T$ equals zero for different indices and $P_i$ for equal indices. Expanding the <finite sum> yields
$$
\overline P_k^2=\sum_{i=1}^k\sum_{j=1}^kP_iP_j=\sum_{i=1}^kP_i=\overline P_k.
$$
Thus \b[every stated partial sum is a <projector>]. Its <image of a linear map> is the <span> of the first $k$ <orthonormal set> of <vectors>; arbitrary sums of <idempotent> <matrices> would not have this conclusion.
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