# Math Help - Help with starting an eigenvalue/eigenvector proof

1. ## Help with starting an eigenvalue/eigenvector proof

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

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

b. Suppose that LG=GL and that L has linearly independent eigenvectors $v_1,v_2,...,v_n$ and G has linearly independent eigenvectors $w_1,w_2,...,w_n$. 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)?

2. Hi davesface.
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.