Show that Axioms 7, 8, and 9 hold. 

I'm having trouble seeing how axiom 7 holds since ku makes the first element a 0 but not kv.. also I'm not sure what the m is in axiom 8 and 9..

The definition of scalar multiplication given in the question is for a general scalar $k$ and vector $\vec u$. And in order to show that the axioms hold, you must use general values, not the specific values given in the question. So $k\vec u = (0, ku_2)$ and $k\vec v = (0, kv_2)$. Then according to the addition defined, $\vec u + \vec v = (u_1 + v_1, u_2 + v_2 \Rightarrow k(\vec u + \vec v) = (0, k(u_2 + v_2))$, and $k\vec u + k\vec v = (0, ku_2) + (0, kv_2) = (0, k(u_2 + v_2)) = k(\vec u + \vec v)$.
For axioms $8$ and $9$, use another scalar $m$.
