Let e be an nth order homogeneous linear difference equation. What does this mean? It means the differences of y are multiplied by various functions of x, and the highest difference has order n, and the right hand side is equal to 0. Here is an example of order 2.
(x3+7x+5)y′′ - Exy′ + sin(x)y = 0
If f(x) is a solution, then we can evaluate f(0), f′(0), f′′(0), and so on, through the first n-1 differences. This builds a vector in n space. Call this map M. Thus M(f) = v, mapping a solution function f to its differences at 0.
Since e is linear and homogeneous, solutions can be added and scaled, hence the set of solutions forms a vector space. Furthermore, M respects addition and scaling. The differences of f+g are the differences of f plus the differences of g. Therefore M is a linear transformation from one vector space into another. The solution space of e has been mapped into n-space.
The previous theorem asserts existence and uniqueness for every vector of initial conditions. This is the inverse of the map described above. Given v, there is a unique f such that M(f) = v. Therefore M is onto (existence), and 1-1 (uniqueness). The set of solutions is isomorphic to n dimensional space.
If the right hand side of e is changed from 0 to r(x), so that e is no longer homogeneous, the solution set is a shifted vector space. Find one solution, h(x), and all solutions are of the form h+f, where f is a solution to the homogeneous equation. The dimension of the solution space is still n; we have simply shifted the entire space by adding h(x) to every function.
All this assumes the range is a field. If the range is an integral domain, embed it in its fraction field, and allow scaling by fractions, and the theorem still holds.