Suppose $\displaystyle L:V\rightarrow V$ and $\displaystyle G:V\rightarrow V$ are linear maps of an n-dimensional space V.

a. Suppose that L and G share the same set of linearly independent eigenvectors $\displaystyle v_1,v_2,...,v_n$ but possibly have different eigenvalues. Show that LG=GL.

b. Suppose that LG=GL and that L has linearly independent eigenvectors $\displaystyle v_1,v_2,...,v_n$ and G has linearly independent eigenvectors $\displaystyle w_1,w_2,...,w_n$. Prove that LG is diagonalizable.

My biggest issue is just with seeing how to start off proofs like this. Are there any general tips (or specific tips to this problem)?