Let $ X = \mathbb{R}^n$ with inner product $⟨x, y⟩ = y^TQx$ for some symmetric positive definite matrix $Q$. I have a few questions about this setup and generally about the kernels.
1- A kernel is the property of linear maps we define on a vector space (right?). So is it correct to say "the associated kernel to $\mathbb{R}^n$"?
2- How do I determine the Kernel and verify $span\{k_1, . . . , k_n\} = X$ and $⟨k_j, k_i⟩ = K_{ij}$. We need to find a map $\phi$ that $K(x,y)= ⟨\phi(x), \phi(y)⟩$ So is $K(x,y)= ⟨x, Qy⟩$ , a kernel?
3- A Kernel $K$ is positive semi definite matrix. if $v^TKv = 0$, can we deduce that $Kv = 0$ for $v \in\mathbb{R}^n$?
4- And if $span\{k_1, . . . , k_n\} = X$ can we assume that can be equipped with an inner product for sure?