Share: Meng Yan's "Understanding Matrices"
Mathematical Calculation
I've mentioned before that I plan to teach myself relativity and quantum mechanics. As the two pillars of modern physics, the mathematics they require is quite "modern" — you can't always rely on the simple methods from high school to do the calculations — so linear algebra is one of the subjects I need to get familiar with. Right now, as a freshman, linear algebra hasn't been offered yet as a course, but my view is: "If you need something, you should go learn it, and you're capable of learning it." Actually, I first came into contact with linear algebra during the summer after my third year of middle school. The textbook I read then, like most other domestic linear algebra textbooks, taught the subject in a way that demanded only memorization and calculation: first introducing determinants via systems of linear equations, then moving on to matrices. Back then I was just memorizing things too — I learned how to compute a determinant, that determinants could be used to solve systems of equations, how matrices are multiplied, and so on. But I had absolutely no idea why — I didn't even understand why the course was called "linear algebra" in the first place. (Of course, it's also possible my math level simply wasn't high enough yet.) Many foreign textbooks teach this very well, in a rigorous and systematic way, but for an average student like me back home, they can come across as too specialized. I had always been hoping to find that sweet spot in between, but unfortunately never did, so I could only feel my way forward through various other channels. more
Meng Yan's "Understanding Matrices" consists of three articles, the earliest of which dates back to 2006. Although he's not some great figure in mathematics, and "Understanding Matrices" isn't exactly a classic work, there's no denying that these three articles played a huge role in helping me understand matrices — and linear algebra more broadly. Through a geometrically intuitive approach, it was the first time I genuinely, clearly understood how to use matrices. Of course, "understood" here doesn't mean I fully grasped everything — it's more that, compared to before, I had that feeling of suddenly seeing the light, of being struck with admiration. So I'd recommend it to anyone interested in linear algebra. While reading about special relativity, I used this same kind of intuitive, matrix-based approach to derive the Lorentz transformation for the first time in a way that actually satisfied me. I believe I'll keep experiencing the subtlety of matrices in physics and geometry going forward, and come to understand them ever more intuitively.
Going forward, I might also write some posts on Scientific Spaces sharing my own understanding of matrices, especially concerning determinants. Comrade Meng Yan has already covered matrices themselves quite thoroughly, and at my current level I don't really have much to add there. But he never got around to determinants, which I have to say is a bit of a shame. When I have time, I'll try to fill in that gap. Meng Yan's blog hasn't been updated in a long while — once I've built up enough of my own understanding, maybe I'll follow in his footsteps and write "Understanding Matrices (Part Four)"...
Meng Yan's blog: http://blog.csdn.net/myan
Download: 《理解矩阵》 by Meng Yan.doc
Translated automatically with claude-sonnet-5; all equations are reproduced verbatim from the source. Copyright remains with the original author.