Assume you mean P3 since that is your example. The standard basis for P3 is and, applying L to each of those,

A basis for is

[tex]\{M1= \begin{bmatrix}1 & 0 \\ 0 & 0 \end{bmatrix},M2= \begin{bmatrix}0 & 1 \\ 0 & 0 \end{bmatrix}, M3=\begin{bmatrix}0 & 0 \\ 1 & 0\end{bmatrix}, M4=\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}

s

so the first column is .

Yes, I get the same thing you do.