Show that the range spaces and null spaces of powers of linear maps t: V → V form descending
V ⊇ R(t) ⊇ R(t2) ⊇ . . .
and ascending
{} ⊆ N (t) ⊆ N (t2) ⊆ . . .
chains. Also show that if k is such that R(tk) = R(tk+1) then all following range spaces are equal. R(tk) = R(tk+1) = R(tk+2) . . . . Similarly, if N (tk) = N (tk+1) then N (tk) = N (tk+1) = N (tk+2) = . . . .