Find a linear transformation $T\colon \mathbb{R}^2 \to P_2(\mathbb{R})$ such that $T(1,2) = 3 + 4x^2$ and $T(3,5) = 1 - 2x + 3x^2$.
I was thinking of approaching the problem the following way:
$ T \left( \begin{matrix} 1 \\ 2 \end{matrix} \right) = \left( \begin{matrix} 3 \\ 0 \\ 4 \end{matrix} \right) $
$ T \left( \begin{matrix} 3 \\ 5 \end{matrix} \right) = \left( \begin{matrix} 1 \\ -2 \\ 3 \end{matrix} \right) $
$ \text{Let } T = \left( \begin{matrix}A & B \\ C & D \\ E & F \end{matrix} \right) \text{ then } $
$ \begin{align*} A + 2B = 3\\ C + 2B = 0\\ E + 2F = 4\\ 3A = 5B = 1\\ 3C + 5D = -2\\ 3E + 5F = 3 \end{align*} $
But converting this system of equations into a matrix and performing row reduction gave me
$ T = \left( \begin{matrix} -13 & 8\\-16 & 46/5 \\-14&9 \end{matrix}\right) $
which when checked against the two given untransformed vectors do not give me the right result.
In addition, the question asks to prove that this linear transformation T is unique. How do I show that a transformation is indeed unique?