I can't seem to wrap my head around the following proof.
I want to show that the linear kernel is a kernel because its Gram matrix is positive semi-definite.
There is plenty of information on the internet, but I don't understand the last step:
linear kernel: $k(x,x')=<x,x>=\sum_{a=0}^N x_ax_a'$
$$\sum_{i,j} c_i c_j k(x_i,x_j) \ge 0$$
$$\sum_{i,j} c_i c_j k(x_i,x_j) = \sum_{i,j} c_i c_j <x_i,x_j> =\sum_{i,j} c_i c_j \sum_{a=0}^N x_ax_a'=\sum_{i,j} \sum_{a=0}^N c_i x_ac_jx_a' $$
So far so good. But now, why can I say that this is equal to:
$$||\sum_{i} \sum_{a=0}^N c_i x_a||^2 \ge 0$$
The $\ge$ is clear just why is the formula transformed in that way?
A similar approach for the dot-product is shown here: click
Thanks a lot!