Equivalent Conditions of Projection Map I got a problem in doing the following:
Let $A_1,\dots,A_k$ be linear operators on a vector space $V$ with dimension $n<+\infty$ such that
$$A_1+\cdots+A_k=I.$$
Prove that the following conditions are equivalent:
1) the operator $A_i$ are projections, i.e. $A_i^2=A_i$.
2) $A_iA_j=0$ for $i\neq j$
3) $rank(A_1)+\cdots+rank(A_k)=n$
I have proved 2) implies 1) and 1) implies 3), but I got difficulty in proving 3) implies 2). It will be good if someone can provide me a hints on that.

Let $V_k$ be the range of $A_k$. Suppose (3) holds. Then  the map $(V_1\oplus \dots\oplus V_k)\to V$ given by $(x_1,\dots,x_k)\mapsto x_1+\dots +x_k$ is a surjective map between spaces of the same dimension; thus, the map is an isomorphism. 
So, every element of $  V$ has a unique representation as a sum of elements of $V_i$, $i=1,\dots,k$. For any $x\in $, the vector $ A_j x$ can be represented in the above form either as   $ A_j x$, or as $\sum_{i} A_i A_j x$. The uniqueness gives you (1) and (2) at once.
