Question:
Suppose T: $P_{2}(R) \rightarrow R^{3}$ is a linear map with matrix ${\big[T\big]}^{\beta}_{\alpha} = \begin{bmatrix} 1 & 0 & -2 \\ -3 & 1 & 4 \\ 2 & -3 & 4\\ \end{bmatrix}$ with respect to the standard bases $\alpha = \{1, x, x^{2}\}$ of $P_{2}(R)$ and $\beta = \{(1,0,0), (0,1,0),(0,0,1)\}$ of $R^{3}$. Find ${\big[T^{-1}\big]}^{\alpha}_{\beta}$
Context:
This question was given as one of the assigned questions for our course but I got it wrong with the feedback "cannot do change of basis here"
My attempt was:
Computing inverse matrix $\big[{T^{-1}\big]}^{\beta}_{\alpha}$ = $\begin{bmatrix} 8 & 3 & 1 \\ 10 & 4 & 1 \\ 7/2 & 3/2 & 1/2\\ \end{bmatrix}$
Computing $\big[{T^{-1}\big]}^{\alpha}_{\beta}$ by doing $\big[{T^{-1}\big]}^{\alpha}_{\beta} = (\big[{I\big]}^{\beta}_{\alpha})^{-1}\big[{T^{-1}\big]}^{\beta}_{\alpha}\big[{I\big]}^{\beta}_{\alpha} =\big[{I\big]}^{\alpha}_{\beta}\big[{T^{-1}\big]}^{\beta}_{\alpha}\big[{I\big]}^{\beta}_{\alpha}$
But I now know that a change of basis is not the way to go about this question. Perhaps there is some flaw in my understanding and I'm not hitting the nail in understanding such problems. What would be the appropriate method to solving it?
Hint:
Note that if $T$ is a map from $P_2(\mathbb{R})$ to $\mathbb{R}^3$ than $T^{-1}$ is a map from $\mathbb{R}^3$ to $P_2(\mathbb{R})$. So the notation for the reresentation of $T^{-1}$ from the basis $\beta$ of $\mathbb{R}^3$ to the basis $\alpha$ of $P_2(\mathbb{R})$ is $[T^{-1}]_\beta ^ \alpha$