Checking if a kernel is valid
Your confusion is that you're using $\phi$ to denote two different mappings. The first one maps its input $v$ to the vector $\begin{pmatrix} v &amp; \mathbf{1} \end{pmatrix}$, and the second one maps it to $\begin{pmatrix} \mathbf{1} &amp; v \end{pmatrix}$. In the same way we can prove that $-1$ is the square of a real number: $-1 = x \cdot x = (-1) \cdot (+1)$.