Let $T: P_2(\mathbb{F}_4) \rightarrow \mathbb{F}_4^2$ be given by: $T(eX^2+fX+g)=(e+\alpha f, (1+\alpha)g)$, where $\mathbb{F}_4 = \{0,1,\alpha, 1 + \alpha\}$ with $\alpha^2+\alpha+1=0$. Find with justification: a) $[T]_A^B$, where $A$ and $B$ are standard bases for $P_2(\mathbb{F}_4)$ and $\mathbb{F}_4^2$, respectively, b) a complete list of $ker(T)$ and $range(T)$.
So far I have plugged in values from A into the function and have a matrix looking like this: \begin{bmatrix} 0 & 0 & \alpha & 1+\alpha \\ 0 & 1+\alpha & 0 & 1+\alpha \end{bmatrix}
Is this enough for part a? From here I should be able to get part b.
Let me know if I am on the right track or if I am doing something wrong. Thanks in advance.