If $KerT$ is a subset of $Sp\{(1,0,1), (0,1,0)\}$ then $T(1,0,1)=0$?

Question

Is it correct that if $Ker(T) \subseteq Sp\{(1,0,1), (0,1,0)\}$

Then $T(1,0,1) = 0$ and $T(0,1,0)=0$?

Or is the only thing that one can be sure of is that $T(\lambda_1(1,0,1) + \lambda_2(0,1,0))=0$ for some $\lambda_1, \lambda_2 \in F$