Define: $T: P_2(R) \rightarrow F $ by $T(A) = Tr(A)$
Compute $T(A)_\alpha^\beta$
Where $\alpha = \left \{\begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix},\begin{bmatrix} 0&1 \\ 0&0 \end{bmatrix} \begin{bmatrix} 0&0 \\ 1&0 \end{bmatrix},\begin{bmatrix} 0&0 \\ 0&1 \end{bmatrix} \right \}$ and $\beta = \left \{ 1 \right \}$
I have no idea where to begin: $T(A) = T(\left(\begin{bmatrix} a_{11} &a_{12}\\ a_{21} &a_{22}\end{bmatrix} \right ) = a_{11}a_{22} $ $$T(\left(\begin{bmatrix} 1 &0\\ 0 &0\end{bmatrix} \right )=1(0),$$ $$T(\left(\begin{bmatrix} 0 &1\\ 0 &0\end{bmatrix} \right )=(0)(0)$$ $$T(\left(\begin{bmatrix} 0 &0\\ 1 &0\end{bmatrix} \right )=(0)(0)$$ $$T(\left(\begin{bmatrix} 0 &0\\ 0 &1\end{bmatrix} \right )=1(0)$$
But then I just get a column of zeros... Can someone help!