To show that T is one one iff T' is one one

Question

Question is to show that T is one one iff T' is one one,where $T:V\to W $and $T':W'\to V'$. I prove one way. Say $T'g_{1}=T'g_{2}$ gives $(g_{1}-g_{2})T=0$ and since T is one one ,it can't be 0 so $g_{1}=g_{2}$. I got confused while dealing other part. Say $T\alpha_{1}=T\alpha_{2}$ gives $(T'g)(\alpha_{1}-\alpha_{2})$=0 ,where g is linear functional in W'.But from this step how can I conclude that $\alpha_{1}=\alpha_{2}$.