Insights on coset in abstract algebra I am taking course on abstract algebra. I found the concept of coset is very important. However, I have not found any insights on my textbook (and by myself). I believe there must be great example on why coset is important in linear algebra(like solution to equations) or some other field of math. Could you provide any examples to connect coset to important theory or application?

I am NOT looking for what will appear on a standard textbook.
A coset can be thought about as a copy of a group, when interpreted as a subgroup of a larger group. So anywhere that groups occur, and their subgroups are interesting at all, we expect to see cosets.

(Note that the cosets of a normal subgroup are of course much more interesting than the cosets of a generic group, because of the group structure on the set of cosets. So these arise in applications much more often.)

Geometry provides many rich examples: here are two that come to my mind immediately: An affine space is a coset, for $G$ a vector space and $H$ a subspace. The orientation of a transformation is a coset where $G$ is $O(n)$ and $H$ is $SO(n)$.
Quotient groups and quotient spaces are enormously important constructions in mathematics.