 Solution

In the terminology used for positive-definite kernels, positive definiteness means positive semidefiniteness of every finite kernel matrix: $k$ is symmetric and, for every $m$, points $x_1,\dots ,x_m$ and real $c_1,\dots ,c_m$,The linear kernel is symmetric, andThus it is a positive-semidefinite kernel, with the identity as its feature map. It need not be strictly positive definite: a zero vector or linearly dependent points can make this quadratic form zero for nonzero $c$. This distinction is essential when interpreting the terminology.