Exercise 4.2.8

The projections of $(1,1)$ onto $(1,0)$ and $(1,2)$ are $p_1=(1,0)$ and $p_2=\frac{3}{5}(1,2)$. Then $p_1+p_2\ne b$. The sum of projections is not a projection onto the space spanned by $(1,0)$ and $(1,2)$ because those vectors are not orthogonal.