= 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 $x_1,\ldots,x_m$ and real $c_1,\ldots,c_m$,
$$
\sum_{i,j=1}^mc_ic_jk(x_i,x_j)\geq0.
$$
The linear kernel is symmetric, and
$$
\boxed{\sum_{i,j=1}^mc_ic_jx_i^Tx_j=\left\lVert\sum_{i=1}^mc_ix_i\right\rVert_2^2\geq0.}
$$
Thus 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.
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