# Representations of direct sums of matrix algebras [closed]

I'm reading Introduction to Representation Theory by Pavel Etingof et al. 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 first 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?!

-

## closed as unclear what you're asking by Julian Kuelshammer, Normal Human, Batominovski, anomaly, user26857Aug 17 at 7:53

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, itâ€™s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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