Answer
See below
Work Step by Step
Assume $v_1,v_2 \in V$
$(T_1+T_2)(v_1+v_2)\\
=T(v_1+v_2)+T_2(v_1+v_2)\\
=(T_2+T_1)(v_1)+(T_1+T_2)(v_2)\\
=T(a_1,b_1,c_1)+T(a_2,b_2,c_2)$
$(T_1+T_2)(cv_1)\\
=T_1(cv_1)+T_2(cv_2)\\
=cT_1(v_1)+ct_2(v_2)\\
=c(T_1(v_1)+T_2(v_2))\\
=c (T_1+T_2)(v_1)$
Hence, $T_1+T_2$ is a linear transformation
Obtain: $Ker(T_1+T_2)=\{v \in V:(T_1+T_2)(v)=0\}\\
=\{v \in V:T(v_1)+T_2(v)=0\}\\
=\{v \in V:T_1(v)=-T_2(v)\}\\
\rightarrow v\in Ker(T_1)
\rightarrow T_2(v)=0$
then $v \in Ker(T_1+T_2)$
