Linear Transformation Proof (Show $L_1(v_i)=L_2(v_i)$)

Question

I understand that the properties of scalar multiplication and addition allow for the expansion of $L_1(v)$ and $L_2(v)$ but I dont see how they are equal. They would only be equal if $L_1 = L_2$, but I don't see how.

Answer 1

When you expand $L_1(v)$ and $L_2(v)$ you still do not know if they are equal. But $$L_1(v) = \alpha_1L_1(v_1)+\dots+\alpha_nL_1(v_n) = \alpha_1L_2(v_1)+\dots+ \alpha_nL_2(v_n) = L_2(v)$$ for any $v$. Then $L_1 = L_2$.