Introductory courses in linear algebra tend to start with either vectors or systems of equations. Which do you prefer to start your course with and why?

“Hello everyone! This is an introductory course in Linear Algebra. Linear Algebra is absolutely fundamental for both the theory and application of mathematics, and it provides an excellent opportunity for us to experience the power and beauty of the abstract approach.

The main actors in Linear Algebra are vector spaces and linear transformations. Vector spaces arise whenever you have things which you can add and scale, and linear transformations are those transformations that respect adding and scaling.

I know this sounds very abstract, and we will of course make it precise and clear. What I want us to understand first, though, is why this abstraction is a good thing.

You see, it’s possible to go through an entire Linear Algebra course and talk about just one type of vectors, usually “tuples” of real numbers such as (23,-6.41,\pi,19). There’s still a lot of techniques and algorithms to learn, and this can be useful for many practical purposes.

But it’s missing out on a fantastic opportunity to widen our horizons and make this course mean so much more. When you work with the abstract definitions, you invest just a little bit additional effort, and get a thousandfold more in return.

Bit strings used in communication and encryption are vectors. Solutions of differential equations are vectors. Quantum states in physics are vectors. Computer images are vectors. Elements of field extensions are vectors. Points in space are vectors. Forces and velocities and electric fields are vectors. Lots and lots of different things are vectors, vector spaces, linear transformations, eigenvalues and so on.

To make any of these work, you must free yourself from thinking that vectors are a bunch of numbers and the only thing you can do with them is multiply them by matrices. You need vector spaces over finite fields, you need infinite dimensional vector spaces, you need to see that laplacians and gradients and Fourier transforms are linear transformations. Then you see why Linear Algebra is so important.

Furthermore, the whole idea of “abstract approach”, of defining entities using axioms and exploring them with observations and proofs, is a tremendously valuable intellectual challenge which is in some ways more important than Linear Algebra itself. This is how real math is done, and at some point you’re going to learn it, and now is a really good time to do it, in the context of Linear Algebra.

So, that’s what we are going to do. We will learn to handle abstract definitions, we will learn to read and create and write proofs, we will learn about fields and vectors and linear transformations and eigenvalues in their most general and wide-ranging manifestations – and we will also learn a few techniques for solving equations and diagonalizing matrices, because the practice of Linear Algebra is important as well.

So, shall we begin?”

(Me, addressing the classroom, imaginary universe #17.)