The other responders above (below?)have given several excellent examples of how more general algebraic principles can be used to clarify linear algebra,but there's one I'm surprised no one's brought up. Actually,that's not entirely true-several HAVE mentioned it-they just couched it in different terms then the one I have in mind.
Consider a group action on a set:
Def: For any group G and X is a set then $\phi : G \times X \to X$ and such that $\phi(e,x)=x$ and $\phi(g,\phi(h,x))=\phi(gh,x) $ for every $x \in X$ and every $g,h \in G$. Then $\phi$ is called a group action on X.
Then consider the definition of an R-module over a ring R.
Def: Suppose that R is a ring and 1$ \in$ R is its multiplicative identity. A left R-module M consists of an Abelian group (M, +) and an operation ⋅ : R × M → M such that for all r, s in R and x, y in M, we have:
1 )r$\cdot$(x+y)=r$\cdot$ x+ r$\cdot$y
3) (rs)$\cdot$x=r$\cdot$ (s$\cdot$x)
4) 1 $\cdot$ x=x
(A right module is defined similarly.)
Looking carefully at this definition, we notice that if we rewrite the scalar action as $L_r$ so that L$_r$(x) = r ⋅ x, and L for the map that takes each r to its corresponding map $L_r$,then (1) states that every $L_r$ is a group homomorphism of M by compatibility.
Also, (2)-(4) assert that the map L : R → End(M) given by r ↦ $L_r$ is a ring homomorphism from R to the endomorphism ring End(M). But this means every left (right) module is a ring action on an Abelian group!
Therefore,every vector space can be thought of as a group action in which the Abelian structure of the field of scalars "acts" on the Abelian group of vectors via multiplication.
Another observation worth mentioning is that if R is a field and G is a group, then a group representation of G is a left module over the group ring R[G].Representation theory is a major branch of abstract algebra with enormous utility in many areas of both pure and applied mathematics where the structure of a group can be analyzed by specific group actions on the a given vector space.