I have been stuck on a problem for awhile now and am in need of some help

Let $\displaystyle T:R^2 \rightarrow R^3$ be defined by $\displaystyle T(a_{1},a_{2}) = (a_{1}-a_{2},a_{1},2a_{1}+a_{2})$. Let $\displaystyle \beta$ be th standard ordered basis for $\displaystyle R^2$ and $\displaystyle \gamma$ = {(1,1,0),(0,1,1),(2,2,3)}. Compute $\displaystyle [T]^{\gamma}_{\beta}$.

Based on the answer in the back of the book it's supposed to be:

$\displaystyle [T]^{\gamma}_{\beta} = \begin{pmatrix} -1/3 & -1 \\ 0 & 1 \\ 2/3 & 0 \end{pmatrix}$

I have absolutely no idea how they got that solution, since when I input the values of the standard ordered basis I get $\displaystyle \begin{pmatrix} 1 & -1 \\ 1 & 0 \\ 2 & 1 \end{pmatrix}$.