I've been trying to get an understanding of the Kernel of image of matrices. I'm studying them in college right now, but the problem is, while I can find a ton of resources on how to find them given a matrix by following steps, I haven't been able to find anything that explains what they are intuitively. Also why have them in the first place?
I hope someone who understands these concepts better can help me.
On
Well they exist because they kind of define the output given the input or if there is a unique solution of an inverse problem etc.
They both defines sets. Kernel defines a set of vectors such that take one of them and multiply it with your vector, you end up with 0 vector. Image on the other hand defines a "target set". Take any vector that can be multiplied with your matrix and multiply them. You cannot end up with a vector that does not belong to the image of your matrix.
The matrix corresponds to a linear function. Generally speaking, for a function $f:X\to Y$, the kernel of $f$ is $f^{-1}(0)$ and the image is $f(X)$.
Here $Y$ is a group, ring, field, vector space etc so it has a zero element $0\in Y$.
Note the kernel $f^{-1}(0)$ is a subspace (subgroup, etc) of $X$ and the image $f(X)$ is a subspace of $Y$.
There is a nice sequence of maps $$0\to\ker f\to X\to \operatorname{im} f\to Y.$$