# Representations of direct sums of matrix algebras

i'm reading Introduction to representation theory by Pavel Etingof etc. (look at http://arxiv.org/pdf/0901.0827v5.pdf) and i want to do most of the exercises. But i must stop by the exercise on page 25. I can prove part (a) easily by direct proof and also by contradiction. I also can prove the fisrt part of the hint: $V=E_{11}V\oplus E_{22}V\oplus\cdots\oplus E_{dd}V$, but than i don't know what to do to finish the exercise. Can someone help me?!

-
So which of the following that you were not able to do? (1) $\Phi_i$ is an isomorphism (2) $S(v)$ a subrep of $V$ (3) $S(v)$ isom to $k^d$ (4) $v\in S(v)$ (5) $V=S(v_1) \oplus \cdots \oplus S(v_k)$ with $v_1,\ldots, v_k$ basis of $E_11 V$. –  Aaron Dec 17 '12 at 21:42