Hey, I can't seem to get anywhere with this big question so if someone could work through this question with me so I understand that would be greatly appreciated.

a) If $\displaystyle g \in M_n$ give a counterexample to $\displaystyle g(x+y)=g(x)+g(y)$

b) Prove $\displaystyle g(\frac{1}{a}x_1 +...+ \frac{1}{a}x_a)=\frac{1}{a}g(x_1)+...+\frac{1}{a}g (x_a)$

c) Using part b, prove that if $\displaystyle G \le M_n$ is finite then $\displaystyle G$ fixes at least one element $\displaystyle k \in R^n$

I think for part c that if $\displaystyle |G|=a$, I can then pick some $\displaystyle x \in R^n$ and consider $\displaystyle x_i = h_ix$ for each element $\displaystyle h_1,...,h_a \in G$.

But I can't get any of part a,b or c anyway.

Thanks guys for any help you can give me.