What is the kernel of a linear recurrence?

Question

There is the Mathematica command: $$ \texttt{LinearRecurrence[ker,init,n]: gives the sequence of length n}\\ \texttt{obtained by iterating the linear recurrence with kernel ker}\\ \texttt{starting with initial values init.} $$ For example, the Pell numbers are defined by $P_0=0$, $P_1=1$, and $P_n = 2P_{n-1}+P_{n-2}$ for $n\geqslant 2$. These are given by the command e.g.: $$\texttt{LinearRecurrence[{2, 1}, {0, 1}, 10]}.$$ So the "kernel" of this recurrence is $(2,1)$. Does this mean that the kernel of any recurrence $P_n = \sum_{j=1}^k a_jP_{n-j}$ is $(a_1,a_2,\ldots,a_k)$? Are there linear recurrences with kernel of some other form? Does this "kernel" have any relation to the concept of a kernel in linear algebra or group theory?