# Vector space stuff

1. Oct 2, 2007

### FunkyDwarf

1. The problem statement, all variables and given/known data
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

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

3. 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 sorta 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

2. Oct 2, 2007

### HallsofIvy

Staff Emeritus
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.