There is a surprisingly versatile part of mathematics that deals with groupings and collections of things. The main ideas involved are really quite easy and practical. If you can make sense of a shopping list written by a spouse or parent, or if you can make sense of the paper cash register slip that the checkout person puts in your bag, you can make sense of this part of mathematics. It is really too bad that the way textbooks and schools talk about this part of mathematics tends to abstract out all the sense-making parts and then hides even further behind fancy but rather forbidding names like vectors and matrices. The part of mathematics I’m referring to is usually called vector algebra, or matrix algebra, or linear algebra.
I want to do a series of blog posts aimed at resurrecting the natural simplicity of the concept of a vector, by avoiding the straitjacket of the standard vector notation(s) for quite a while. Of course I’ll connect up the work presented here with the standard notations and terminology, but there are other useful notations – I like a very flexible notation that is more ‘verbose’ than the standard one and that allows you to see what is going on. This flexible notation looks more like a cash register receipt or like the shopping cart from Amazon.com or like a database table.
Traditionally, we first encounter vectors in physics class, where velocity and force are said to have a direction, not just an amount. We learn to draw an arrow with a length equal to the amount of force, and with a direction that’s the direction in which the force is applied, and we learn to call this arrow a vector. At this time we may even learn about the combined effect of two forces acting together but each in a different direction (e.g. a model airplane held by a string). When we later learn in math class about vectors, what we learn may feel completely different: we may now be told that a vector is a sequence of numbers, written in this form: (3,2,10,1) and you may be shown rules for adding, subtracting and multiplying these sequences without having any idea where these rules came from. If you are shown these completely arbitrary-looking rules, you may not even realize that something strange is going on – for this may not look any different from other parts of your secondary school math education. When they say that (3,2,10,1) + (1,2,3,4) equals (4,5,13,5) this may appear innocent enough, and when they say that (3,2,10,1)-(1,2,3,4) equals (2,0,7,-3) this may not look too strange either, but if you then were to guess that (3,2,10,1) times (1,2,3,4) would be (3,6,30,4) you may hear instead: “no, (3,2,10,1) times (1,2,3,4) equals 41, and this is called an inner product.” The teacher may show you how the inner product is computed, but the odds are that from the description you will still have no idea as to what real-life question “41” is the answer to, let alone why you should care.
My promise to you is that this series will be easy and fun and sensible, ; that by the end, you’ll see what the big deal is, and why the textbook people think it is so important, and you will get to see how the standard way of presenting the material loses everything that made it easy, fun and sensible to start with but ended up with an extremely compact notation. At that point you will be in a very good place to judge if the trade-off was worth it.