In Silverman's Arithmetic of Elliptic Curves page 93, he defines $$\det:E[m]\to \mathbb{Z}/m\mathbb{Z}, \quad \det(aT_1+bT_2,cT_1+dT_2)=ad-bc$$ where $\{T_1,T_2\}$ is a basis for $E[m]$.
I don't get why elements in $E[m]$ are of the form $(aT_1+bT_2,cT_1+dT_2)$ and not just $aT_1+bT_2$.