Prove a function is a linear transformation

Question

I have to prove that $T : M_{2x2} \rightarrow P_{3}$ defined below is a linear transformation:

$T \bigg(\bigg( \begin{matrix}a & b \\ c & d\end{matrix}\bigg)\bigg) = (a+b) +(a-2c)x+(c-d)x^2+(a-c-d)x^3$

I understand that, generally, I need to show that is closed under addition and closed under multiplication, but I am not sure where to start for this problem.

Answer 1

Guide:

For closed under addition, let $A, B \in M_{2 \times 2}$.

• First compute $T(A)$ explicitly as the entries of $A$.
• Compute $T(B)$.
• Compute $A+B$.
• Compute $T(A+B)$
• Compute $T(A)+T(B)$.
• Check if $T(A)+T(B)=T(A+B)$.