Let \(\displaystyle A\in M_{22}\) with \(\displaystyle A=\begin{bmatrix}a&b\\c&d\end{bmatrix}\). What is the most basic mapping \(\displaystyle L\) we can define such that \(\displaystyle L:M_{22}\mapsto P_3\)? (What do elements look like in \(\displaystyle P_3\)?)

That's how you define the mapping! \(\displaystyle L:M_{22}\mapsto P_3\) where \(\displaystyle L\left(\begin{bmatrix}a&b\\c& d\end{bmatrix}\right)=ax^3+bx^2+cx+d\).

Can you justify that this mapping is an isomorphism?

That's how you define the mapping! \(\displaystyle L:M_{22}\mapsto P_3\) where \(\displaystyle L\left(\begin{bmatrix}a&b\\c& d\end{bmatrix}\right)=ax^3+bx^2+cx+d\).

Can you justify that this mapping is an isomorphism?