You should study Linear Algebra.
Linear algebra lies at the heart of mathematics. It allows us to construct and solve systems of linear equations, unifying otherwise disparate topics (read: functional analysis, physics, regression) through a common set of guiding principles and applicable techniques.
A typical introductory course introduces matrices and vectors, finding the four fundamental subspaces of a matrix (check out my article on that topic here), solving systems of linear equations, and modelling problems from related fields, among many other interesting concepts. In this article, we’ll explore why you should start learning linear algebra today.
So, why should you study linear algebra?
From a practical perspective, it’s a topic which transcends mathematics and the physical sciences ― many applied fields, including hot-topic disciplines such as data science and machine learning, all utilise aspects of linear algebra in their practice. Although finding a reformulation isn’t always obvious, there are hundreds of processes and techniques which can be rewritten in a format that’s compatible with a linear algebra approach; this allows us to then apply our mathematical knowledge to unfamiliar situations and more easily produce meaningful results. Linear algebra helps you to solve difficult problems.
Linear algebra: the study of linear equations and linear transformations, and their representation through vectors and matrices.
Who uses linear algebra?: Mathematicians, physicists, engineers, biologists, economists, AI/ML engineers, data scientists, and many more ― all in all, a significant part of the skilled labour market.
Linear algebra also helps you in the modern job market. Machine learning engineers construct AI models in which data and weights are stored as vectors and matrices, opening up an arsenal of matrix operations for model inference; computing a forward pass on a trained model is akin to matrix multiplication, and the backpropagation algorithm stores partial derivatives as a vector. Data scientists use matrices as a convenient data structure for holding thousands or even millions of data points efficiently, and economists need systems of linear equations to model macroeconomics and economic policy.
Linear algebra is everywhere, and studying it will make you stand out among other candidates applying to the same job ― showing that you can work fluidly with underlying theory and self-study content are skills that employers value. The independence, creativity, and attention to detail developed through studying linear algebra are ‘soft skills’ which will augment your CV and profile immensely, just as much as the mathematical techniques you’ll encounter and learn to use.
Mathematical rigour: explicitly stating assumptions, constructing conclusions and theorems from a set of given truths, and proving their truth (or lack thereof) in a comprehensive manner.
Linear algebra makes you a better thinker. In high school, mathematics courses are generally computation-based; students learn techniques and use them with different numbers in questions, with formal examinations following the same format. Rote memorisation and consistency are prioritised over problem solving ability ― a test paper only cares about candidates finding the correct result and demonstrating the ‘right’ method for high marks to be awarded.
Learning linear algebra at a university-level standard is completely different. Most courses emphasise understanding over parroting information, promoting a deep appreciation of mathematics which stands in stark contrast to the dull, and unfortunately all too common, routine of blindly reciting theorems and formulae. We learn to prove, not take for granted, which ultimately improves our ability to solve problems and independent reasoning skills.
Real life problems don’t usually arrive as different numbers packaged into a common format; it’s a luckless fact of life that at one point or another you’ll be expected to work on complex tasks in your job with little supervision (blame corporate work culture for that), and being unprepared for such responsibility is a recipe for disaster. Almost any undergraduate mathematics course would suffice to acquaint you with rigour, but taking a linear algebra course exposes you to developing independent and logically sound thinking without unnecessarily sophisticated mathematics.
There is hardly any theory which is more elementary [than linear algebra], in spite of the fact that generations of professors and textbook writers have obscured its simplicity by preposterous calculations with matrices. ― Jean Alexandre Eugène Dieudonné
There’s a myriad of weird and wonderful module options in a standard mathematics undergraduate degree, but the vast majority are inaccessible to interested readers without having worked through months or even years of prerequisite materials. Linear algebra, on the other hand, is commonly taught as one of the first courses that mathematics students ever take in university. As long as you’re comfortable with high school maths, it’s a great starting point!
Getting started with linear algebra
If you’re now thoroughly convinced that linear algebra is the right course for you, I highly recommend MITOpenCourseWare’s 18.06Sc course for getting to grips with the basics. Gilbert Strang’s style of lecturing is incredibly intuitive and informative, and is also the primary resource from which I first started to learn about this fascinating subject.
Georgia Tech also has a brilliant online interactive linear algebra textbook, replete with a vast amount of exercises; I recommend working through 18.06Sc and supplementing it with as many exercises as you can reasonably work through.
Finally, I also regularly publish articles about linear algebra here on my Medium page; I usually cover topics which I find particularly interesting, and they might be of use to you for building up a more thorough intuition and understanding of foundational theory and certain techniques.
