Finding an unknown linear transformation given that $T(1,1)=(1,0,2)$ and $T(2,3) = (1,-1,4)$

Hint: $\{(1,1),(2,3)\}$ is a basis of $\mathbb R^2$.

I don't think that your reason works. If a linear transformation exists, it should be a 2 by 3 matrix. Let this matrix have the entries $a$ and $b$ on the first row, $c$ and $d$ on the second and $e$ and $f$ on the third. Performing matrix multiplication on <1,1> and <2,3> to get <1,0,2> and <1,-1,4> gives the following systems of equations to solve: $a+b=1$ with $2a+3b=1$, and $c+d=0$ with $2c+3d=-1$ lastly $e+f=2$ with $2e+3f=4$ These systems produce unique values for the matrix' entries. Sorry for my poor formatting