Is the set of all projection matrices a convex set?

Question

The set $\phi=\{P| P^2=P\}$ contains all projection matrix. Is this set $\phi$ convex?

Answer 1

Hint: consider $$ A=\left( \begin{array}{ccc} 1/2 & 1/2 \\ 1/2 & 1/2 \end{array} \right)\\ B=\left( \begin{array}{ccc} 1 & 0 \\ 0 & 0 \end{array} \right)\\ C = 1/2A + 1/2B $$

Answer 2

Hint: In $\mathbb R^2$, consider projection onto the two basis vectors $P_1(x_1,x_2)=(x_1,0),P_2(x_1,x_2)=(0,x_2)$, then $P_1/2+P_2/2=I/2$.