Let $S = \{(x, y) \mid x, y \in R \}$ and define vector addition and scalar multiplication in the same way as $R^2$. Define an inner product in $S$ according to the formula $\left<u, v\right> = 2v_1 u_1 - v_1 u_2 - v_2 u_1 + v_2 u_2$ and show that $S$ equipped with this inner product is a real inner product space.
Now, I know that for $R^2$ we can let $u = (u_1, u_2)$ and $v = (v_1, v_2)$ be vectors in $S$ and that for addition $ (u + v) = (v + u)$ and multiplication $k(u, v) = (ku + kv)$.
Then for inner product spaces I know the 4 axioms have to hold:
$\left<u, v\right> = \left<v, u\right>$
$\left<u + v, w\right> = \left<u, w\right> + \left<v, w\right>$
$\left<ku, v\right> = k\left<u, v\right>$
$\left<v,v\right> > 0$, and $\left<v,v\right> = 0$ iff $v = 0$
But I'm not quite sure what the question is asking for and how to use these findings to answer it. Is it just asking to show the 4 axioms hold according to the defined formula?