Let $\mathcal{X}$ be any space. A symmetric function $k: \mathcal{X} \times \mathcal{X} \to \mathbb{R}$ is called a kernel function if for all $n \geq 1, x_1, x_2,..., x_n \in \mathcal{X}$ and $c_1,...,c_n \in \mathbb{R}$ we have
$$\sum\limits_{i=1}^n \sum\limits_{j=1}^n c_i c_j k(x_i, x_j)\geq 0.$$
I can understand that the n data points $x_1, x_2,..., x_n$ are arbitrary points in the space $\mathcal{X}$. But what about $c_1,...,c_n \in \mathbb{R}$? Are these $c_i$'s fixed (pre-assigned) or any real numbers?
Now we introduce the linear kernel. The trivial kernel on $\mathbb{R}^d$ defined by the standard scalar product:
$$k: \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}, k(x,y)=<x,y> $$
I know it satisfies the symmetry property obviously. But how to show this is satisfied by the positive semi definiteness?