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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?

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3 Answers

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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.)

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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.

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As mentioned by @triple_sec it can be said 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.

These are those vectors that after undergoing a transformation, squished themselves into a zero vector because the determinant = 0. (That means the region squished into 0 volume); These vectors are also called free variables or non-pivot columns;

N(A) = N(RREF(A))

The kernel of a transformation is equal to the kernel of RREF of the transformation. That is the vectors that get squished in an RREF or the original matrix are going to be equal. The pivots help us define the system of equations in reduced forms. The null space helps us define the originality or loyalty of the equations/matrix transformation.

In a non-full-rank matrix-transformation say of 3D to 2D, there is a line fill of vectors that lay at the origin. A 3D to 1D transformation has a plane full of vectors that lay at the origin.

In Ax = [0] When v happens to be a 0 vector, The null space/kernel gives you all possible solutions to all the system of equations.

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