Mathematicians are chronically lost and confused (2014)
I have a great personal story that highlights how long the journey of understanding mathematics is.<p>I took linear algebra my freshman year of college. It was the non-math major course, so it didn&#x27;t require proofs. I got an A+ in the class. Not just an A, an A+. I was able to obtain such a high grade by taking tons of practice tests, and since the actual tests were basically mildly veiled calculations, I just had to map the question to the right calculation. So for instance, if after a little interpretation, I figured out that the question was asking for me to calculate the singular value decomposition of a given matrix, I would mindlessly compute, check my algebra, and move on.<p>However, it was very clear to me by the end of the course that I didn&#x27;t really understand what the heck linear algebra was about.<p>Five years later, I started a job as an algorithmic trader. One of the first things my boss wanted to do was to do a Principal Component Analysis (PCA) of bond price movements. This is a very common thing to do. I didn&#x27;t know what PCA was, but I read a short paper he gave me and I was able to grok it. After reading that paper and actually performing the PCA (which by the way was basically one line of R code), I finally came to understand the core essence of linear algebra, which is the idea of linear transformations. I was able to connect the equation Ax=lambdax to the geometry of what an eigenvector meant. Through a little more reasoning, I realized that every real matrix corresponded to a linear transformation of that space via a rotation, a reflection, a stretching, a shearing, etc. At that point, all of the mindless calculations I had been doing half a decade earlier instantly clicked, and I was enlightened.<p>This was literally half a decade later after I &quot;aced&quot; my linear algebra class. I know that it seems absolutely ridiculous that I could &quot;score so well&quot; in a math class yet so clearly miss the core idea behind the entire class, but that&#x27;s been my experience with math for as long as I can remember. You start by doing the calculations and just getting comfortable with them. Some arbitrary time later, you have an insight and suddenly everything is so crystal clear and trivial that you wonder how you could even <i>not</i> have understood it before.<p>Oh, and even to this day, I don&#x27;t understand what singular values actually are. Something to do with a mapping from the row space to the column space, blah blah. I&#x27;m sure if I spent an hour to read about them and picture the geometry, I could figure it out, but I just haven&#x27;t gotten around to doing it.