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