Hoffman and Kunze Linear Algebra Section 6.8 Decomposition Theorem

Question

Below you can find the related theorem and part of the proof that I am confused about. I understand that with the related definition of $E_i$'s we get k linear operators on $V$that satisfy

1. $E_1+...+E_k=I$, where $I$ denotes the identity operator on $V$
2. $E_iE_j=0$ if $i{\neq}j$
   However, as far as I know, we additionally need to show that each $E_i$ is a projection on $V$ to conclude their ranges can be taken as direct summands of $V$, which can be confirmed by showing $E_i^2=E_i$ for each $i$. It seems the writers claim that from 1 and 2, one can directly conclude that $E_i$'s are projections on $V$, but I cannot see why.

Answer 1

Apply $E_i$ to formula (1) $\sum_j E_j=I$ and using (2) to eliminate all $i\neq j$ using $E_i E_j=0$ gives $E_i^2=E_i$ Thus, those two are sufficient to show the $E_i$ are projections which split up the space.