Exercise 1. Let Mn(R) denote the vector space (actually an R-algebra) of n × n matrices with entries in R. Given A ∈ Mn(R), let HA = {B ∈ Mn(R)| AB = BA}. Show that HA is a subspace of Mn(R).
Exercise 2. Given A ∈ Mn(R), let JA = {f(A)| f(X) ∈ R[X]} ⊆ Mn(R) Show that JA is a subspace of Mn(R) by showing that JA = Span{I, A, A2 , A3 , . . .}.
Exercise 3. In the notation of the preceding two exercises, show that JA ≤ HA.
Exercise 2. Given A ∈ Mn(R), let JA = {f(A)| f(X) ∈ R[X]} ⊆ Mn(R) Show that JA is a subspace of Mn(R) by showing that JA = Span{I, A, A2 , A3 , . . .}.
Exercise 3. In the notation of the preceding two exercises, show that JA ≤ HA.