# Kernel

## Main Question or Discussion Point

I'm working on this to try and find out more on kernels. I've written out basically what the robotic solving methods are for them but I simply don't have answers, help?

Let G denote the set G = {f : R → R | f is inﬁnitely diﬀerentiable at every point x ∈ R}. (R as in the reals)
(a) Prove that G is a group under addition. Is G a group under multiplication? Why or why not?
(b) Consider the function ϕ : G → G deﬁned by ϕ(f) = f′
Prove that ϕ is a homomorphism with respect to the group operation of addition. What is the kernel of ϕ?
(c) Consider the function ψ : G → G deﬁned by ψ(f) = f′′ − f. Prove that ψ is a homomorphism with respect to the group operation of addition. What is the kernel of ψ?

My work:

For (a) we need to prove that G satisfies all the group axioms under addition. Let f, g, h be infinitely differentiable at every point x in R

Closure : f + g is infinitely differentiable at every point x in R
Associativity : (f + g) + h = f + (g + h)
Identity : Find a function e in G such that for all functions f in G such that f + e = e + f = f
Inverse : For all f in G find an inverse ~f in G such that f + ~f = e

Is G a group under multiplication?

For (b) and (c)

The kernel of a homomophism are all elements of the domain group that map to the zero element, e, of the codomain.

What is the zero element of the codomain? This is e I found for G in the group axioms. In this problem e is the zero function zero(x) = 0?

So for (b), the problem is asking for which functions f in G does phi(f) = zero, i.e. which functions has f'(x) = 0 for all x in R?

for (c), the problem is asking for which functions f in G does psi(f) = zero, i.e. which functions has f''(x) - f(x) = 0 for all x in R?

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I'm working on this to try and find out more on kernels. I've written out basically what the robotic solving methods are for them but I simply don't have answers, help?

Let G denote the set G = {f : R → R | f is inﬁnitely diﬀerentiable at every point x ∈ R}. (R as in the reals)
(a) Prove that G is a group under addition. Is G a group under multiplication? Why or why not?
(b) Consider the function ϕ : G → G deﬁned by ϕ(f) = f′
Prove that ϕ is a homomorphism with respect to the group operation of addition. What is the kernel of ϕ?
(c) Consider the function ψ : G → G deﬁned by ψ(f) = f′′ − f. Prove that ψ is a homomorphism with respect to the group operation of addition. What is the kernel of ψ?

My work:

For (a) we need to prove that G satisfies all the group axioms under addition. Let f, g, h be infinitely differentiable at every point x in R

Closure : f + g is infinitely differentiable at every point x in R
Associativity : (f + g) + h = f + (g + h)
Identity : Find a function e in G such that for all functions f in G such that f + e = e + f = f
Inverse : For all f in G find an inverse ~f in G such that f + ~f = e

Is G a group under multiplication?
No, since we may have zeros ##z## of a function ##f##, which cannot be inverted ##1 \neq g(z)\cdot f(z)## regardless how we define ##g##.
For (b) and (c)

The kernel of a homomophism are all elements of the domain group that map to the zero element, e, of the codomain.

What is the zero element of the codomain? This is e I found for G in the group axioms. In this problem e is the zero function zero(x) = 0?

So for (b), the problem is asking for which functions f in G does phi(f) = zero, i.e. which functions has f'(x) = 0 for all x in R?
Yes. And ##phi(f)=f'=0## means that ##f## is a constant function: ##f(x)=c \in \mathbb{R}## for all ##x\in \mathbb{R}\,.##
for (c), the problem is asking for which functions f in G does psi(f) = zero, i.e. which functions has f''(x) - f(x) = 0 for all x in R?
Yes. And ##f''(x)=f(x)## is true for all functions ##f(x)= \alpha e^x -\beta e^{-x}##.

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