Exercise 38, § 7

Let $K$ be a commutative field of characteristic 0 and $E$ a finite-dimensional vector space over $K$. Let $p_1, \ldots, p_m$ be projectors of $E$ onto vector subspaces of $E$ such that $1_E = p_1 + p_2 + \cdots + p_m$. Show that $E$ is the direct sum of the $p_i(E)$ and that the $p_i$ are pairwise permutable (take the traces of the two sides).