Sum and difference of projections I have been going through the Halmos' Finite dimensional vector space, where in section 42 I got struck at the theorems regarding the combination of projections. I can't understand how the conditions of the linear combination of projections define the conditions for the existence of the projection. In particular,

Suppose $E_1$ and $E_2$ are projections on $U=M_1\oplus N_1$ along $M_1$, and $V=M_2\oplus N_2$ along $M_2$ respectively, and the underlying field has characteristic $\ne2$.
(i) $E_1+E_2$ is a projection iff $E_1E_2=E_2E_1=0$; if this condition is satisfied, then $E_1+E_2$ is the projection on $M$ along $N$, where $M=M_1\oplus M_2$ and $N=N_1 \cap N_2$.
(ii) $E_1-E_2$ is a projection iff $E_1E_2=E_2E_1=E_2$; if this condition is satisfied, then $E_1-E_2$ is the projection on $M$ along $N$, where $M=M_1\cap N_2$ and $N=N_1 \oplus M_2$.

I can't get why the direct sum condition is employed for $M$ and intersection for $N$. It gets more obscure when this condition for $\oplus$ and intersection swaps when we talk about $E_1-E_2$.
I would be really glad if someone could give me some explanation using examples for these theorems. Thanks in advance.
 A: It is always useful to think of these situations in terms of 3-space. As an example think of $E_1$ as the projection on the $z$-axis ($M_1$) along the $xy$-plane ($N_1$), and $E_2$ as the projection on the $x$-axis ($M_2$) along the $yz$-plane ($N_2$). Geometrically, it is easy to see that $E_1+E_2$ is the projection on the $xz$-plane ($M_1\oplus M_2$) along the $y$-axis ($N_1 \cap N_2$). 
Algebraically: $E_1(x,y,z)=(0,0,z)$ and $E_2(x,y,z)=(x,0,0)$, so that we have $(E_1+E_2)(x,y,z)=E_1(x,y,z)+E_2(x,y,z)=(x,0,z)$.
Using the same projections for $E_1$ and $E_2$ above you can verify (ii) as well...
Intuitively projections preserve certain subspaces, and "kills" the remaining subspace of the vector space (that forms a direct sum with the preserved space). Now if two projections preserve different subspaces, adding them together will preserve a larger subspace and "kill" off less. (I hope this makes some sense!). And again you can formulate a similar intuitive description of (ii).

As per OP's request:
ok, you can use the same spaces as above to verify part (ii), but a more suggestive example perhaps is to let $E_3$ to the projection on the $xz$-plane along the $y$-axis. Then $E_3-E_1$ is a projection, and: $(E_3-E_1)(x,y,z)=E_3(x,y,z)-E_1(x,y,z)=(x,0,z)-(0,0,z)=(x,0,0)$. Geometrically, you are projecting onto the $xz$-plane, but then you remove the $z$-coordinate by subtracting and so in effect you are projecting onto the $x$-axis. So the nullspace of $E_3$ is augmented by the projection space of $E_1$... so you can see if the projection space of the subtracted operator does not coincide with the nullspace of of the first operator we have a direct sum of these spaces as the new nullspace. And then naturally the projected space of the combined operator must decrease...hope this helps.
