Solution
= Solution
The <image of a linear map> of $P_1$ is exactly the one-dimensional <span> of $q_1$: its formula always returns a multiple of $q_1$, and $P_1q_1=q_1\ne0$. Hence \b[$\operatorname{rank}(P_1)=1$].
The <rank-nullity theorem> gives $\dim\mathcal N(P_1)=n-1$. Since $\mathcal C(I_n-P_1)=\mathcal N(P_1)$ by (e)(i), \b[$\operatorname{rank}(I_n-P_1)=n-1$]. This includes the case $n=1$, where the complementary <projector> is zero.