Word origin / meaning of 'kernel' in linear algebra

Question

It may be the dumbest question ever asked on math.SE, but...

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Given a real matrix $\mathbf A\in\mathbb R^{m\times n}$, the column space is defined as $$C(\mathbf A) = \{\mathbf A \mathbf x : \mathbf x \in \mathbb{R}^n\} \subseteq \mathbb R^m.$$

It is sometimes called image or range.

• I'm OK with the name 'column space' because $C(\mathbf A)$ is the set of all possible linear combinations of $\mathbf A$'s column vectors.
• I'm OK with the name 'image' because if I consider $\mathbf A \mathbf x$ as a function then $C(\mathbf A)$ is this function's image (the subset of a function's codomain).
• I'm OK with the name 'range' because I can consider $C(\mathbf A)$ as a range of a function $f(\mathbf x) = \mathbf A \mathbf x$.

Unfortunately, I'm not happy with the name kernel. $$\ker(\mathbf A) = \{\mathbf x: \mathbf A\mathbf x = \mathbf 0\}\subseteq \mathbb R^n$$

The kernel is sometimes called null space and I can fairly understand where this name came from -- it's because this set contains all the elements in $\mathbb R^n$ that are mapped to zero by $\mathbf A$.

Then why is it called 'kernel'? Any historic background or colloquial meaning that I completely missed?

Answer 1

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.

Answer 2

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.)