I use linear algebra quite a lot in applications, but I do not have a very strong abstract algebra background (i.e. around the level of an intro course, just knowing the basics of rings, groups, ideals, the first isomorphism theorem).
Of course, the latter is far more general, so I was interested in how it can given insight into the former.
For instance, I thought it was interesting to look at the set of rotations as the group $SO(3)$.
So, essentially I'm curious as to what insights one can glean from "viewing linear algebra though an abstract algebra lens".
Not necessarily practical tools, but rather more important are ones that aid in understanding or intuition (i.e. provide something new).
As a particular example, what are the relations of vector spaces to these abstract structures, and is viewing them from that point of view helpful?
Best Answer
The "Rank-Nullity" Theorem from Linear Algebra can be viewed as a corollary of the First Isomorphism Theorem, which may be more intuitive.
Suppose $T:V\to V$ is a linear transformation. Then by First Isomorphism Theorem, $V/\ker T\cong T(V)$.
So $\dim V-\rm{Null}(T)=\rm{Rank}(T)$, which is the Rank-Nullity Theorem.
This may be more intuitive than the traditional Linear Algebra proof of Rank-Nullity Theorem (see https://en.wikipedia.org/wiki/Rank%E2%80%93nullity_theorem).