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I have a great personal story that highlights how long the journey of understanding mathematics is. I took linear algebra my freshman year of college. It was t
by vecter 10y ago
I have a great personal story that highlights how long the journey of understanding mathematics is.
I took linear algebra my freshman year of college. It was the non-math major course, so it didn'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.
However, it was very clear to me by the end of the course that I didn't really understand what the heck linear algebra was about.
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'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.
This was literally half a decade later after I "aced" my linear algebra class. I know that it seems absolutely ridiculous that I could "score so well" in a math class yet so clearly miss the core idea behind the entire class, but that'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 not have understood it before.
Oh, and even to this day, I don't understand what singular values actually are. Something to do with a mapping from the row space to the column space, blah blah. I'm sure if I spent an hour to read about them and picture the geometry, I could figure it out, but I just haven't gotten around to doing it.
- lordnacho 10y agoWow, that's oddly similar to my story. LinAlg in college, nothing but a blur of matrices. Algo trading for work. Now there's a reason to do it, it makes sense. About SVD, btw, it is another path to PCA, one that solves certain problems that PCA does not. https://jeremykun.com/2016/04/18/singular-value-decomposition-part-1-perspectives-on-linear-algebra/ https://jeremykun.com/2016/04/18/singular-value-decompositio...
- IndianAstronaut 10y agoMath programs in general suck at telling you why a particular technique is useful in the real world, or more importantly, how to model. Lots of ways to solve an existing model, but very few touch the much harder task of building a model in the first place.
- Retra 10y agoStudents and parents are demanding that their education fees generate employable graduates, not creative or intelligent ones. This trend will probably continue as long as education keeps getting more expensive.
- fahadkhan 10y agoSomewhat ashamedly, I have a 1st Class BSc Honours in Mathematics with much the same experience. It's fairly easy to fake an understanding of mathematics if you can understand that the exams paper will be a variant past exam papers.
- kenjackson 10y agoI've had a similar experience several times. I feel like in college I was mentally much faster, but lacked insight (and probably curiosity). There have been several times when I've picked up an old math or CS textbook and read something I never really fully understood in college -- it almost instantly makes intuitive sense to me now -- 20 years later. There's an old saying that college is wasted on the young... in many ways I believe it is true.
- wmsiler 10y agoLinear algebra is the sort of topic where it seems like you can always go deeper. I also got an A in my linear algebra class in college. When I got to grad school for math, I took the first year graduate course on algebra and saw linear algebra in terms of algebraic things like modules and representations. I decided that I hadn't really known linear algebra before, but that I did then. Then I took differential geometry, which involves studying infinite collections of vector spaces parameterizes by points on a manifold. I realized I still hadn't know linear algebra, but after that I certainly did. Then I took a functional analysis class, where we did infinite dimensional linear algebra, where a lot of the finite dimensional theory has analogues but everything is more complicated (e.g. instead of finite bases and dot products, you consider things like Fourier analysis and Hilbert spaces). The same realization that I don't know linear algebra came when taking a Lie algebras class, and again when learning homological algebra, and probably a few other times as well. There are certainly lots of areas of math that I've never explored that take linear algebra in some other direction (for example, I don't have any idea what the applied math guys do with....). It's really an amazingly vast subject, especially considering that it's usually just thought of as a tool used to study more advanced topics.