1. Makélélé and Linear Algebra


inear algebra is the Claude Makélélé of science and mathematics. Makélélé is a well-known, retired football player, a French international. He played in the famous Real Madrid team of the early 2000s. That team was full of “galácticos” — the most famous and glamorous players of their generation. Players like Zidane, Figo, Ronaldo and Roberto Carlos. Makélélé was hardly ever in the spotlight, he was paid less than his more celebrated colleagues and was frequently criticised by fans and journalists. His style of playing wasn’t glamorous. To the casual fan, there wasn’t much to get excited about: he didn’t score goals, he played boring, unimaginative, short sideways passes, he hardly ever featured in match highlights. In 2003 he signed for Chelsea for relatively little money, and many Madrid fans cheered. But their team started losing matches.

The importance of Makélélé’s role was difficult to appreciate for the non-specialist. But football insiders regularly described him as the work-horse, the engine room, the battery of the team. He sat deep in midfield, was always in the right place to disrupt opposition attacks, recovered possession, and got the ball out quickly to his teammates, turning defence into attack. Without Makélélé, the galácticos didn’t look quite so galactic.

Similarly, linear algebra does not get very much time in the spotlight. But many  galáctico subjects of modern scientific research: e.g. artificial intelligence and machine learningcontrol theory, solving systems of differential equations, computer graphics, “big data”, and even quantum computing have a dirty secret: their engine rooms are powered by linear algebra.

Linear algebra is not very glamorous. It is normally taught to science undergraduates in their first year, to prepare for the more exciting stuff ahead. It is background knowledge. Everyone has to learn what a matrix is, and how to add and multiply matrices. The question “what is a matrix?” is typically answered in a particularly boring way: a matrix is a double array of numbers. For example

\left(\begin{array}{ccc}3 & 2 & 5 \\ 1 & 0 & 0 \end{array}\right)

is an example of a 2×3 matrix, a rectangular collection of numbers with 2 rows and 3 columns. I think that giving this answer is somewhat similar to first year humanities undergraduates being told that the definition of the word “poetry” is “collection of words”. Why are rectangular collections of numbers so important in science?

In this blog we will explore a very different way of understanding linear algebra, where the standard concepts such as matrices, vector spaces and bases will feature very rarely. We will also touch on applications of linear algebra in this new light.

Makélélé’s reputation was completely rehabilitated after a fantastic few years at Chelsea and the corresponding failures of Madrid after his departure. It’s time for linear algebra to have its time in the spotlight.

Continue reading with Episode 2: Methodology, Handwaving and Diagrams.