Let R and C be the field of real numbers and the field of complex numbers, respectively.
(1) Determine whether the set $\mathbb{R}^2 = \{(x, y) : x, y \in \mathbb{R}\}$ forms a vector space over $\mathbb{R}$
under the operations of addition and scalar multiplication are given as follows:
$(x, y) + (z, w) = (x + z, y + w)$ for all $x, u, z, w \in \mathbb{R}$,
$\alpha(x, y) = (y, \alpha x)$ for all $\alpha, x, y \in \mathbb{R}$.
Justify your answer.
[6 Marks]
