Definition of kernel

Question

Is the kernel the same as the union of all level sets (or perhaps the set of all level sets)?

Kernel (https://en.m.wikipedia.org/wiki/Kernel_(set_theory)): $\left\{\, \left\{\, w \in X \mid f(x)=f(w) \,\right\} \mid x \in X \,\right\}$

Level set (https://en.m.wikipedia.org/wiki/Level_set) $L_c(f) = \left\{ (x_1, \cdots, x_n) \, \mid \, f(x_1, \cdots, x_n) = c \right\}~,$

Why is the kernel described as the vectors that are mapped to zero in vector spaces? Surely there are more level sets than that?

score 1 · Answer 1 · answered Mar 17 '18 at 07:59

1

The kernel of a linear map $f$ is a level set, yes. And it is a particulary important one, since $\ker f=\{0\}\iff f$ is injective.

answered Mar 17 '18 at 07:59

José Carlos Santos

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Is the usage not the same as in the set theory kernel? – Emil Mar 17 '18 at 08:01
@Emil Not at all. – José Carlos Santos Mar 17 '18 at 08:06
It feels like it could be the kernel in the set theory sense with some additional constraints, but I can't formulate it properly, tried to say it in terms of linear homomorphisms but failed. – Emil Mar 17 '18 at 08:41

score 1 · Accepted Answer · answered Mar 17 '18 at 11:06

The kernel for general maps between sets is an equivalence relation: if $f\colon X\to Y$, then the kernel is the equivalence relation $\sim_f$ defined by $$ a\sim_f b\text{ if and only if }f(a)=f(b) $$ The Wikipedia page identifies this relation with the partition induced by it: $$ \bigl\{ \{w\in X:f(x)=f(w)\}:x\in X \bigr\} $$ where, for $x\in X$, $\{w\in X:f(x)=f(w)\}=\{w\in X:x\sim_f w\}$ is the equivalence class (or level set) of $x$.

When linear maps are concerned, there's a better description, because when $X$ and $Y$ are vector spaces and $f$ is linear, $$ f(a)=f(b)\text{ if and only if }f(a-b)=0 $$ so the kernel can be described just by the vector subspace $N=\{x\in X:f(x)=0\}$.

There's no real difference, except that in vector spaces (but also in groups or rings) the description of the kernel is handier.

The key fact is that $\sim_f$ is more than an equivalence relation: it is a congruence, that is, an equivalence relation that preserves the operations on the structure: if $a\sim_f b$ and $a'\sim_f b'$, then $$ a+a'\sim_f b+b' $$ and, when $\gamma$ is a scalar, also $\gamma a\sim_f \gamma b$.

The fact that a vector space (and likewise a group or a ring) has an operation which is associative, with neutral element and inverses, allows for the simpler description in terms of a single subset rather than with a partition.

I think I need to find a proof of why there are not more level sets for linear maps to feel full understanding. But I guess it is not surprising that only linear maps are considered when talking about kernels for vector spaces, rather than any function. — Emil, Mar 17 '18 at 14:54
@Emil You can describe the kernel both ways; the fact is that, for additive groups (in particular for vector spaces) the level sets are known once you know the equivalence class of $0$. — egreg, Mar 17 '18 at 14:57

Definition of kernel

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