1. Show that for the polynomial kernel function $K: \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}$ defined as
$K(x, y) = (x^T y + 1)^2$
can be expressed as
$K(x, y) = \phi(x)^T \phi(y)$,
where $\phi: \mathbb{R}^2 \to \mathbb{R}^6$ is defined by:
$\phi(x) = \begin{pmatrix} 1 \ \sqrt{2}x_1 \\sqrt{2}x_2 \\ x_1^2 \\ x_2^2 \\ \sqrt{2}x_1 x_2 \end{pmatrix}.$
