Sum of projections. Vector space

Question

Let $V$ be a vector space over $F$. If $E_1$ and $E_2$ are projections onto independent subspaces of $V$, then $E_1+E_2$ is a projection over $V$?

Answer 1

If you define a projection as a linear map $P$ such that $P\circ P=P$, then the answer is negative. Consider the maps $E_1,E_2\colon F^2\longrightarrow F^2$ defined by$$E_1(x,y)=(x,0)\text{ and }E_2(x,y)=\left(\frac{x+y}2,\frac{x+y}2\right).$$Both of them are projections, but their sum isn't.