Answer
See below
Work Step by Step
Since
$T(A)=AB-BA$
we have
$$T(A_1+A_2)=(A_1+A_2)B-B(A_1+A_2)\\
=A_1B+A_2B-BA_1-BA_2\\
=(A_1B-BA_1)+(A_2B-BA_2)\\
=TA_1+TA_2$$
$$T(cA)=cA_1B-BcA_1\\
=cA_1B-cBA_1\\
=cT(A_1)$$
Thus it is a linear transformation.