Show that each eigenspace of an n n matrix A is an invariant subspace, as defined

Question:

Show that each eigenspace of an n × n matrix A is an invariant subspace, as defined in Exercise 7.4.32.
Exercise 7.4.32
The subspace W of a vector space V is said to be an invariant subspace under the linear transformation L: V → V if L[w] ∈ W whenever w ∈ W. Prove that ker L and mg L are both invariant subspaces.