Proof Vector Space of Shift Maps is Isomorphic to R2

[FunkyDwarf]

Homework Statement

Show that the space of all shift maps is indeed a vector space over R and that there is a linear bijection between it and R2

Homework Equations

10 Axioms of vector spaces
Definition of bijection (1-1, onto)
For 1-1: f(a) = f(b) -> a = b.

The Attempt at a Solution

Ok ignoring the vector space proof for the moment my main problem was defining this space to begin with. I sort of saw it as the set of functions f st f(x) = x + a where x and a are sets or matrices of values from the field R. The only problem here is there is no limit really to the dimension of this space and so getting it to be a bijection to R2 could be a problem (here i assume that isomorphisms have the same dimension) or am i to limit our function space to dimension 2?

Im kinda muddeled on this one guys
Cheers
-G

 

[HallsofIvy]

surely the problem said more than that? Didn't it say "all shift maps on R²"? That's the only way that last part could be true.