Proof about Commuting Matrices and Invariant Subspaces

[Math_Fortnite_Lover]

I will be honest, this question I have no clue where to even start. I know what eigenspace or gen eigen space or invariant space means but don't even know line 1 of proof. If someone can point me in the right direction or show me some ideas that would be fantastic.

 

[blamocur]

To show that $E_\lambda$ is an invariant subspace for B one needs to show that $\forall x \in E_\lambda : Bx \in E_\lambda$. From the definition of $E_\lambda$ it is enough to prove that if $Ax=\lambda x$ then $ABX=\lambda BX$. But $ABX=BAX = B\lambda x = \lambda BX$. QED.