Here's a hint to get you started for a. : The eigenvectors constitute a basis of V. So it suffices to show that LG and GL have the same value on each of these eigenvectors.
Suppose and are linear maps of an n-dimensional space V.
a. Suppose that L and G share the same set of linearly independent eigenvectors but possibly have different eigenvalues. Show that LG=GL.
b. Suppose that LG=GL and that L has linearly independent eigenvectors and G has linearly independent eigenvectors . 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)?