I'm currently studying linear algebra with a great deal of attention to its definitions and theorems, but most of the time it seems that the motivation is always in applied or computational mathematics.
But I've seems comments like
(...) mathematics nowadays is basically the study of turning problems into linear algebra and solving them.
that got me thinking:
How researchers utilize linear algebra? Which areas benefit most from it? Which areas do not?
On
Speaking based on a four-decade research career in a variety of fields, I can say that linear algebra is extremely useful math in a wide range of applied disciplines, especially in anything to do with machine learning (which affects nearly all applied fields), and big data. The underlying reason is that linear algebra deals with large numbers of equations and variables simultaneously... as do ever more real-world problems. As the OP quote states, many problems consist of casting them into a form of linear algebra and then using optimization routines for solving them.
(In the other window on my computer, I'm using linear algebra to infer the direction of illumination based on the pattern of lightness on the outer boundary of the girl's face in Johannes Vermeer's famous painting Girl with a pearl earring... Who would have guessed that?!)
True, some of the most abstract and fundamental research mathematics does not rely heavily upon linear algebra (number theory, topology, etc.), and if that is your direction, then linear algebra may not be essential for you. There are a number of deep research-level problems in linear algebra, but surely not as many--and not as deep--as the highly abstract, deep problems elsewhere in math. I don't know when the last Abel Prize or Fields Medal was awarded for work in linear algebra, but I don't suspect it was recently.
But for the vast majority of researchers who use math, linear algebra is extremely useful and powerful.
On
Linear algebra is used heavily in topology, it is the basis of homology and cohomology theory: you have lots of linear maps, their images and kernels, quotients thereof, etc. etc. Computationally you have to do row and column reduction to convert matrices to Smith normal form, in order to figure out the ranks of the homology/cohomology groups, so lots of linear algebra stuff there.
And, if you like new-fangled applications, homology groups (and thus linear algebra) are all over the place in the theory of persistent homology which is designed for applications to big data.
Linear Algebra is really used everywhere, throughout pure mathematics and applied. It's an extremely fundamental subject and in no way should you skimp on it. In fact, it's one of the classes you should make sure to master the most. I seriously cannot think of an area of math which doesn't use LA (except perhaps logic/set theory, but maybe even those).
Addressing your particular question
Linear Algebra is part of algebra, and most books on algebra will assume you have taken one or two classes in Linear Algebra.
The some of main objects of study in Algebra are groups, rings, and modules. Linear algebra concepts are used for every one of them, but in particular, modules. Modules themselves are a generalization of vector spaces, so if you study modules (very big area of study) you are studying a generalized linear algebra.