Word origin / meaning of 'kernel' in linear algebra

Solution 1:

The word kernel means “seed,” “core” in nontechnical language (etymologically: it's the diminutive of corn). If you imagine it geometrically, the origin is the center, sort of, of a Euclidean space. It can be conceived of as the kernel of the space. You can rationalize the nomenclature by saying that the kernel of a matrix consists of those vectors of the domain space that are mapped into the center (i.e., the origin) of the range space.

I think a somewhat analogous rationale might motivate the designation “core” in cooperative game theory: It denotes a particular set that is of central interest. (In this case, it denotes—loosely speaking—the set of such allocations among a given number of persons that cannot be overturned by collusion among some of them. This property lends the core a sense of stability and equilibrium, which is why it is so interesting.)

Solution 2:

The imagery is consistent with inhomogeneous equations $Ax = b$ where the degrees of freedom in the answer are those of $Ax = 0$ and the latter could be seen as the invariant core of the problem separate from the particularities of different $b$ (for some values there are solutions, for others there can be no solutions).

Whether this really was the historical origin I cannot say. Of course it makes sense for group homomorphisms.