GoKawiilDisclaimer: no AI was used to write this. Any errors, awkward sentences, and weird tangents are 100% organic, free-range, and human-made.
How to actually learn anything?
For a long time I have always been asking myself what it really means to learn or grasp something in math. I think this can be extended to other areas, but math is very often particularly distant from daily ordinary life when it comes to many of its concepts, and it’s hard for me (at least) to have an intuitive and natural understanding of it — the type of understanding that gives you some sort of foundational structure of the concepts and allows you to just reconstruct its details by yourself later in the future, with none or limited consultation. I have experienced a few times this feeling of having really captured some sort of general structure in some areas, that allows me to express myself or deduce facts only consulting what I currently understand about it. One good example is my first experience with Real Analysis. At first, it looked very confusing and difficult, but after a while you capture some sort of structure of the main concepts and you can mostly run the concepts yourself.
But one thing took me too long to understand: traditional math books very often fail to give you a path that allows you to capture structure, or to grasp concepts in a more intuitive or natural way (that would allow you to reconstruct it later). This is because most math books present math in its final formalized version. But they hide the path that was taken to get there: often messy, experimental, trial-and-error and tentative.
This blog assumes you already have basic knowledge in math. It’s just not practical to teach everything from the ground up, but my goal is to write here some of the connections that helped me grasp concepts.
Starting Somewhere
I’ve chosen to start from Linear Algebra (L.A). It’s one of the most applicable and accessible areas in math. When I was introduced to a central Linear Algebra concept called Singular Value Decomposition (SVD), it took me time to really understand what it is, and I think most books fail to explain it in a simple way simply because they usually start from the final conclusion formally stated, which makes no sense to the reader. I think it’s critical to introduce motivation on why this was created in the first place. Most things in applied math came from a specific problem someone was trying to solve. Linear Algebra is a great place to start not only because it’s somehow simpler than more advanced non-linear areas, but also because it’s connected to so many other areas, such as calculus, information theory, image processing, machine learning, and many others. It’s some sort of basic or central area, although I think no matter where you start, if you continue to dig enough, you will reach connections to other areas — and in L.A this is even more evident.
Let’s arrive naturally at SVD without even aiming at it. The matrix of a Linear Transformation A \mathbf{A} A can be written differently for the same L.T depending on the chosen input and output bases. To see that, take a deliberately simple linear map — stretch the x x x-axis by 3, leave y y y alone:
A ( x , y ) = ( 3 x , y ) \mathbf{A}(x, y) = (3x,\ y) A ( x , y ) = ( 3 x , y )
In the standard basis, its matrix just reads off where the basis vectors land:
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