Establish criteria for f in the Range(L)

[mathgirl313]

So..here's the problem:

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).

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).

Then from what I was told I want to look at what x(t) looks like for x = 0,1,2,...5 and determine what needs to be true of f. But that's all the further I can get, any advice?

 

[gtoguy97]

For f to be in the Range(L), for each t (t=0,1,2,3,4) the value of x(t+1) must be equal to Ax(t)+f(t). So, for example, if t=0, then x(1) = Ax(0)+f(0). And for t=4, then x(5) = Ax(4)+f(4). Therefore, for f to be in the Range(L), we need to have a set of equations that are satisfied by the values of f(0), f(1), .., f(4). These equations will involve the elements of the matrix A and the values of x(0), x(1), ..., x(5).