Machine Learning – Test Quadratic Kernel

We have the given quadratic core.

K (x, y) = (xT y) ^ 2

φ ([x1, x2]) = {x1x1, x1x2, x2x1, x2x2}

Show that K (x, y) = φ (x) T φ (y) for vectors of arbitrary length n.

I can show that this works for a two-dimensional vector, but I'm confused about how to test it for a n-dimensional vector.