ODE, Establish criteria for f in Range(L)

[mathgirl313]

Homework Statement

Let Z={x: {0,1,..,5} → ℝ^n} (column vector with real entries) and define T:Z→ℝ^n by Tz = z(0) - z(5). Let X = Ker(T) and let Y={y: {0,1,..,4}→ℝ^n}. Define L:X→Y by
(Lx)(t) = x(t+1) - Ax(t) where A is an invertible matrix.

Establish criteria for f in Range(L).

Homework Equations

The Attempt at a Solution

I know for f to be in Range(L) there exists x in X such that L(x) = f
so (Lx)(t) = f(t) for all t,
then x(t+1) - Ax(t) = f(t) for all t,
or x(t+1) = Ax(t) + f(t).

 

[Nino MMP]

I am not sure how to continue or if this is the correct approach to solve the problem. Any help would be appreciated.