Why do polynomial sequences have the coefficient $\sqrt2$ in front of them as in $\phi(x) = [x_1, x_2, \sqrt2x_1x_2, x_1^2, x_2^2]$

Question

Why do polynomial sequences have the coefficient $\sqrt2$ in front of them as in $\phi(x) = [x_1, x_2, x_1x_2, x_1^2, x_2^2]$?

For example if our original feature space is:

$$x = [x_1, x_2]$$

Then why do we have $\sqrt2$ in the following polynomial expansion:

$$\phi(x) = [x_1, x_2, \sqrt2 x_1x_2, x_1^2, x_2^2]$$

?

Answer 1

This has to do with the kernel trick. To elucidate, one can deconstruct a kernel $\kappa$ into an inner product of (possibly finite dimensional) feature vectors $\phi^T\phi$ such that $\kappa = \phi^T\phi$.

Now can you see why the $\sqrt{2}$ coefficient arises in the feature vector?