Chapter 1
We Start Here

This book is a story about linear algebra, the main character of which is the star of this discipline, the vector. We will start by defining this concept, that prides itself on its simplicity. But don’t mistake simplicity for lack of power; from an uncomplicated vector we will arrive at complex methods like the single value decomposition and the principal component analysis.

My journey began when I was only four years old, and my father gave me my first book on equations. Since then, I have never looked back. Mathematics flowed in my mind, and calculations came out as naturally as a delicate butterfly landing on a ravishing red petal of this miracle of nature that we so often call a flower… don’t be scared already! We are just at the second paragraph, and this is not true. I am just a regular guy who was most likely kicking a ball around when he was four. But, being a typical fellow, my struggle with mathematics was real during a specific time in my life, my first couple of years at university. This was because of a combination of a bad attitude and a need for content to be structured more like a story than a manual. I was scared of equations and blamed everything I could, except myself, for my lack of success in understanding mathematics. When I look back now, I can see that it is impossible to understand anything with that attitude.

Symbols and Greek letters are the alphabets of mathematics, whereas equations are the words that represent abstract concepts. One needs to try to understand how to read this syntax, as it will bring significant benefits in the future. Unfortunately, mathematics has no sound, so I don’t think you can expect good results by using a hands-on approach where you learn by ignoring the syntax, as you might do with a musical instrument. Still, as a mathematician, I can’t say that this way is not possible. After all, the realm of uncertainty is where we do our best work. Once I overcame this first hurdle and I started to be able to read equations, another issue arose. I knew concepts in isolation, but relating them to one another seemed impossible. Different books have distinct structures and expose the same ideas in varying sequences, which became another obstacle for me. Now I say that I was lucky, but at the time, I considered myself the unluckiest person in the world. I could not have been more wrong.

The itinerary whereby I began putting concepts together and understanding mathematics started on the day I missed the meeting where we, the students, were due to meet the professors who would be supervising our university theses. I can’t provide a good reason for missing this meeting that won’t make you think I am an idiot, but hey, sometimes things have a funny way of resolving themselves.

When I finally returned to the mathematics department, my colleagues came to me with a concerned look, enquired where I had been, and told me that I was in trouble as I had landed the worst supervisor ever. This lady was famous for being extremely demanding and challenging to get along with. On that same day, the path of my life changed completely. Indeed, she was demanding, and she presented me with a project I knew very little about, but had to master. She made me study, and did not give anything back to me unless she saw that I had made an effort. I had to go back to basics, but this time I decided to start with the most elementary concept of each subject, then I studied it in such a way that everything moving forward would have to be the result of knowledge I had previously acquired. This way, I could put everything into context.

I am still a data scientist. Well, in reality, I am a mathematician. I don’t like that job title, but I also need to pay the bills. It helps me. The point is that my Master’s thesis was the hardest thing I have ever done, and the conclusion is that if you make a significant effort to learn the basics, what comes afterwards will be a smoother ride. There is a lot of talk these days about the wealth gap, but I feel that another gap is emerging, one in knowledge. We like to press buttons and obsess about whatever is the next exciting thing. Modern entertainment and social media have given us all attention deficit disorder. When this is associated with a right-now mentality, it significantly contributes to this problem.

If I go back to my first experience with a mathematics book, I can understand why this might happen. With so much information out there, the minimal hurdle presented to somebody trying to learn something new is enough to make them try something else. There are a lot of us pressing buttons. Still, only a few of us are building them. If you want to succeed as a data scientist, it would be better to take a button-builder path. What this means is that you will have to learn mathematics.

I wrote this book aiming to help the reader to start and never have to look or go anywhere else for further information. There will be no need for notebooks, pens, laptops, or pencils: just the safe blueprint, a mask, and the machine gun. Oh sorry, those last items might have come from the “bank project” list… actually, you won’t need much more than the Pythagorean theorem: my mistake.