We know that a lack of understanding feels like chaos, muddled thinking and an uncomfortable lack of confidence about some concept. In short, it is the inability to debug our brain and take it from a question to an answer. Let us call this not having understanding driven by “first principles”. It often leads to a subtle but critical problem: being wrong and not being able to spot it. The anecdote to this state of chaos is learning from first principles.
I’ve often had this experience: certain technical topics look extremely difficult to break into. No matter how many standard resources and popular content that you read through, every one of them seems to speak a foreign language. I’ve felt this while studying topics like complex numbers, Fourier’s transform, relativity, to name a few. I am of the opinion that either I’m too dumb or writers of many popular expositions on such topics have not understood it clearly.
Click here for introduction to this series and motivation. Differential Equations An equation is about balancing two sides of a (weight) balance scale. The equations that we study in school (x + 4 = 9) are about finding the number which will balance the scale. But there could be equations for which solution which balances things is a function and not a number! Specifically, you could have equations which relate a change in certain quantity with respect to that quantity.
Click here for introduction to this series and motivation. Linear Algebra Basics There are two ways to look at matrices. One view is about a matrix representing some data (to be covered in later part of this series) Another view is: matrix encodes the operation of a linear system on its inputs. In this (3) view, a matrix represents a linear map. Matrix-vector multiplication is taking a vector represented in one coordinate system and transforming it to vector in another co-ordinate systems.
I had to study a lot of maths in higher secondary and engineering. There were 2 maths subjects for 2 years after secondary school and 3 maths subjects in the first two years of engineering. Despite studying so much maths, graduate school was a rude shock for me. I just couldn’t wrap my head around of a lot of lectures. The problem was the way maths was taught. Most of the teachers taught maths as an end in itself, mostly devoid of the most important question: “why”?