TITLE: Insights into linear algebra from abstract algebra
QUESTION [44 upvotes]: 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?

REPLY [3 votes]: I think circulant matrices provide a good example of crossroads between linear algebra and abstract algebra. 
For a first approach, see paragraph "Analytic Interpretation" in the Wikipedia article about circulant matrices: (https://en.wikipedia.org/wiki/Circulant_matrix)
Let us take the case of  $3 \times 3$ circulant matrices that can be decomposed in the following way:
$$\begin{pmatrix}a&b&c\\c&a&b\\b&c&a\end{pmatrix}=a\underbrace{\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}_{I_3=C^0}+b\underbrace{\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}}_{C=C^1}+c\underbrace{\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}}_{C^2=C^T}$$
Thus any circulant matrix is a a matrix polynomial in $C$ (as introduced in formula above).
Moreover, it is connected with harmonic analysis, a major branch of abstract algebra, because it is diagonalized by Fourier matrix $F_n$:
Let us take an example ; let $\omega=e^{i 2\pi/3}$ be a primitive cubic root of unity.
Let us consider matrices : 
$$C=\begin{pmatrix}1&2&4\\4&1&2\\2&4&1\end{pmatrix} \ \ \& \ \ F_3=\tfrac{1}{\sqrt{3}}\begin{pmatrix}1&1&1\\1&\omega&\omega^2\\1&\omega^2&\omega\end{pmatrix} \ \ \& \ \ D=\begin{pmatrix}7&0&0\\0&-2+\sqrt{3}i&0\\0&0&-2-\sqrt{3}i\end{pmatrix}$$
With these matrices, one can write:
$$C=F_3DF_3^{-1}$$
Moreover the diagonal elements of $D$ constitute the DFT (Discrete Fourier Transform) of the first column of $C$. (see for example : (https://math.stackexchange.com/q/2629712)).
As a corollary (because of the correspondence between product and cyclic convolution $\star$ by Fourier Transform), denoting by $C_V$ the cyclic matrix associated with first  $V$, we have :
$$C_V \times C_{V'}=C_{V \star V'}.$$