I’ve been learning Linear Algebra for five months now. I effectively started at the very beginning of 2026, and have chewed on a piece of it nearly every day since. I’ve watched Gilbert Strang’s lectures, done the reading in his textbook, followed the recitations, done exercises, referenced other textbooks, and relied heavily on my AI tutor. It’s been almost psychedelic. And here I am coming out of the trip, not yet sure how to verbalize what I’ve learned, but sure it’s been impactful.
I’ve just finished Singular Value Decomposition (SVD). I get the general idea of SVD as a means of translating large matrices into two rank-one vectors. The example used is image processing and projection. When we watch visual media, what is before us is a large matrix of colors, a substantial amount of data, that is compressed and communicated using single value decomposition.
Like recent topics, I feel like the returns on my time and effort watching lectures, reading the corresponding chapters, masticating the examples, and so forth, have greatly diminished. Linear Algebra is abstract. It’s been so from the starting blocks. But it’s grown increasingly so as the course has progressed. Abstract topics build on top of previous abstract topics, theorems on theorems, all communicate through notation jargon. Take for instance the following SVD proof
This demonstrates the v vectors are orthonormal eigenvectors of the product of A-transpose and A. It does, right? Take the Greek aside for a moment. To understand this, you need to first understand:
- Eigenvector : A nonzero vector whose direction remains unchanged when a matrix acts on it, being scaled only by an eigenvalue. And what’s an eigenvalue? The scaling factor by which a matrix grows or shrinks and eigenvector when a matrix acts upon it.
- Orthonormal Eigenvector: An eigenvector that has unit length and is perpendicular to all other eigenvectors in an orthonormal set. And what does perpendicular mean? Why is it significant?
- Matrix Transpose: A matrix formed by swapping the rows and columns of another matrix.
And you’ll need to understand what U and V matrices are in this context, and that capital Sigma is a diagonal matrix of sigmas (lower case) that each represent the length of the columns of matrix A. And you must understand how matrix multiplication works. And so on. And to understand this you must keep all of this in working memory, at least until it’s hard-coded in your brain sufficiently to understand it intrinsically.
I understand that this is how abstract ideas are communicated. Maybe how they must be communicated. Over time, surely this becomes more commonplace. But I’m not there yet, not where it doesn’t overheat my brain in a way that no other study has. I spend hours processing topics, and hours more just trying to understand what practice questions are asking for.
Strang, although no doubt an amazing mathematician, can be difficult to follow. His lectures often seem to drop right into ideas abruptly, assuming understanding in the realm of his understanding. “So, take this obvious truth…and so it would be this matrix, which means what if it’s inverted-transposed in the left-nullspace orthogonal but not orthonormal? Of course it means this”. And he dashes about the room, scribbling hither and thither on the blackboard, sometimes without coherently sequencing themes in a manner that mere plebs like myself can understand. And his book, well that can at time be like reading a manual in Martian.
I don’t want you to get it confused – I think Linear Algebra. The study of linear transformations, the mapping of vectors to other vectors in a fashion that preserves the core functions of those vectors. A framework for looking at how patterns interact with each other and change each other’s behavior. It is a field that can be applied to phenomena all around us. Although I am often confounded by the mechanics of Linear Algebra, I see how powerful it is. I definitely will need another round with it to process it better. I’d also like to get my hands dirty using it in real applications, particularly in Economics and Computing, given those are where this study is pointing.
Having mentioned my grander pursuits, it’s important to stress how important my time is. Growing older, the value of my time is very evident. Although I enjoy learning for the sake of learning, I am aspiring to reach a point where I can confidently march into the DEDP Micromasters program and kick ass. Ideally, I’d like to start in September. I still have my Calculus 3 course to finish and need to study Probability and Statistics. WIth that in mind, I’ve chosen to call the Linear Algebra class ‘a wrap’. It’s not without reluctance that I do so, particularly as I’m just short 3 sections from completion:
- Linear Transformations and their Matrices
- Change of Basis; Image Compression
- Left and Right Inverses; Pseudoinverses
However, at this point I feel like continuing would be more performative than substantive. As such, I’m at peace with giving it a break. I’m sure this won’t be the last I see of Linear Algebra.
I’m not going to expound upon deeper takeaways from Linear Algebra right now. I want to let it all set in, and then later revisit it to think more on it. I’ll definitely be back. I’ve been extraordinarily challenged by Linear Algebra, and there is much yet to learn. So, until later, keep your eigenvectors pointed in the right direction.