This problem and the one in your book are the same - the solution for R^3 is generalisable to any field F of dimension 3 (hence F^3).
Show that the vectors (1,1,0), (1,0,1) and (0,1,1) generate F^3.
I found an example that resembles this problem in the book, only F refers to a field (I'm pretty sure). The problem in the book is "the vectors (1,1,0), (1,0,1), and (0,1,1) generate R^3 since an arbitrary vector (a1, a2, a3) in R^3 is a linear combination of the three given vectors; in fact, the scalars r, s, and t for which
r(1,1,0) + s(1,0,1) + t(0,1,1) = (a1, a2, a3)
What's the difference between these two problems? Are they solved the same way?