Importance of the kernel

Question

I have taken linear algebra as a sophmore in college, and I know there are some relations to the kernel with regard to transforms of functions. I want to know what significance it yields to the seasoned mathematician who knows how important it is. I'm more interested with understanding the kernel in a function/functional sense.

Answer 1

Well, a linear map $f$ is injective if and only if $\ker f=\{0\}$. Besides, we can use the kernel to compute the dimension of the range of $f$. By the rank-nullity theorem, if the domain of $f$ is $V$, then $\dim f(V)=\dim V-\dim\ker f$.

Answer 2

In the introduction to linear algebra the kernel seems underwhelming. That subspace that maps to 0... I want to learn about the interesting things that linear maps can do! Sending things to 0 seems a real bore.

Some ways of thinking about the kernel that you might not be aware of:

If you have a linear map with a non-trivial kernel, information is getting compressed and the kernel tells you what gets compressed.

You might have a geometric projection. You are mapping from $\mathbb R^3 \to \mathbb R^2$ the kernel tells you which vector you are projecting along.

If you have a system of equations with a space of solutions. If you can find one particular solution, then add that point to the vectors that span the kernel to find the space of solutions.

The homogeneous ordinary differential equations are linear maps! Solving the diff eq. amounts to finding the kernel of that map.

Linear Algebra is the jumping off point for abstract algebra, and the kernel of a group homomorphism is not just a sub-group but a normal subgroup....

Answer 3

One of the most obvious comments to make here is that the kernel tells you not just about the pre-image of the zero vector but about the pre-image of any vector in the range of the transformation. This is because the general solution to $T(x) = y$ is just a translation of the kernel of $T$. i.e. The general solution to $T(x) = y$ can always be expressed in the form $a + \ker{T}$ where $a$ is a single vector satisfying $T(x) = y$. To put it more simply, every solution to $T(x) = y$ is of the form $a + n$, where $T(a) = y$ and $n \in \ker(T)$. Thinking along these lines we start to see that `except' for the kernel, a linear transformation is invertible... like given $y$, there is a unique $a$ for which the solution set is $a + \mathrm{ker}(T)$

This eventually leads to what is called the "first isomorphism theorem" of vector spaces. It is really a fundamental way of thinking in algebra. The vector space version can be made very explicit and simple but there are 1st isomorphism theorems for groups, rings, modules and other more complicated algebraic objects.

Answer 4

The kernel also leads to the higher level concept of quotients. From a colloquial point of view a quotient can be thought of as gluing objects together in a way that perserves the original structure (a congruence to be precise).

For a linear transformation $T: V \to V$ a the kernel tells you exactly how you glue things together for all points (because of linearity) and gives the vector space isomorphism

$V/\ker T \cong Im T$

So the vector space you get by gluing together elements of $V$ if they are both in $\ker T$ is isomorphic to the image of $T$ (also called the column space, $T(V)$ etc). This result is called the first isomorphism theorem and appears in many guises all over mathematics.