2x2 Matrix to Polynomial Transformation

This problem comes from Section 2.4, Number 4, of Linear Algebra by Friedberg, Insel, and Spence.

Problem:

$\displaystyle T: M_{2\times2}(R) \to P_2(R)$ where $\displaystyle T\left(\begin{array}{cc}a&b\\c&d\end{array}\right) = (a + b) + (2d) x + (b) x^2$.

The bases for $\displaystyle M_{2\times2}(R)$ and $\displaystyle P_2(R)$ are, respectively, given as:

$\displaystyle \beta = \{ \left(\begin{array}{cc}1&0\\0&0\end{array}\right) , \left(\begin{array}{cc}0&1\\0&0\end{array}\right) , \left(\begin{array}{cc}0&0\\1&0\end{array}\right) , \left(\begin{array}{cc}0&0\\0&1\end{array}\right) \}$ and $\displaystyle \gamma = \{ (1), (x), (x^2) \}. $

We are asked to compute the matrix form of the transformation, $\displaystyle [ T ]^\gamma_\beta $

Solution: I transformed each matrix, but this gave me 4 coefficient vectors, which means I end up with a 3x4 matrix. That means the input vector would have to be 4x1. Is that correct? How come the input isn't a 2x2 matrix?

EDIT again: the matrix I ended up with was:

$\displaystyle [ T ]^\gamma_\beta = \begin{pmatrix}\1&1&0&0\\0&0&0&2\\0&1&0&0\end{pmat rix}\right) $

EDIT: hit submit, rather than preview

Re: 2x2 Matrix to Polynomial Transformation

NVM - I got it. Was thinking about how a basis composed of matrices works. Not sure how to say it in words, but each component of the 4x1 matrix stands for how many of each of the basis matrices there are.