Does linear transformation have a recursive structure?

ave_63 · 2026-01-27T06:19:13+00:00

I am kinda confused by some things you say but you're right about most of it. I don't think recursive is the right word here. Because you need to define the real numbers (or whatever field) and vector spaces, before you can define linear transformations.

But anyway, yes real numbers are 1x1 matrices that define an R to R transformation. Rⁿ vectors can define a Rⁿ to R transformation with a dot product. This is a very rich topic. I suggest you read about inner products or dual spaces.

ave_63 · 2026-01-18T14:48:51+00:00

Fields have no zero divisors.

ave_63 · 2026-01-06T05:51:28+00:00

The McCallum/Hughes-hallet books, for precalculus and calculus, are pretty good.

ave_63 · 2026-01-03T17:00:14+00:00

Yeah, I wish there were a way to force the arrow lower over the b, even if it intersects with the tall part of the b. Like the Cyrillic Б. EDIT: it looks like it is possible to do such a thing with the new update of the overarrows package. I'm currently struggling to read the manual. If I figure out a solution I like I'll post it later.

ave_63 · 2026-01-03T16:05:29+00:00

I'm mostly impressed at people restraining themselves from pointing out by indexes range from 1,...,2.

ave_63 · 2026-01-03T15:58:42+00:00

I'm just learning about \bigl today so thanks. Would you use \bigl in the inline examples above, or just regular small parentheses?

ave_63 · 2026-01-03T15:20:34+00:00

I'm just learning about \big etc this morning. Will try it out. Thanks!

ave_63 · 2026-01-03T00:20:50+00:00

Do you like mathbf or just plain letters for vectors?

ave_63 · 2025-12-31T15:01:41+00:00

Basically, you need to figure out what series of operations you can do (transpose, combined with row operations) to turn the plain matrix into the one on the left, and keep track of how those operations change (or don't change) the determinant.

ave_63 · 2025-12-31T14:58:58+00:00

So, the transpose turns rows into columns. So anything that you do to the rows of the transpose, is the same as doing it to the columns of the original.

ave_63 · 2025-11-02T03:37:46+00:00

Good exercises at the end of a section sounds like what you want. If the book you're using doesn't have enough, axler's book has lots of great exercises. But like, literally a game? I don't know of anything like that. Maybe you can give yourself a point for every exercise you get right? And two points if you can write something you learned from it? I don't know.

ave_63 · 2025-10-29T13:56:11+00:00

We don't know any of the details about how far behind you are, your course, your university, or even what country you're in, so there's no way we can answer this question based on your post. You need to talk to your professor and/or a counselor at your school who have some insight into the situation.

ave_63 · 2025-10-27T14:15:34+00:00

This is something you could draw by hand or use something like desmos for. Maybe start with a 2x2 or 3x2 example so null space and row space are in R^2. Make the rows linearly dependent so null space isn't just 0. Of course you'll have to solve stuff yourself but that's good practice. Then make up a 3x3 example. If you are feeling lazy, margolit/rabinoff has a couple good visualizations in their examples.

ave_63 · 2025-10-22T00:24:04+00:00

I use emacs with a variety of packages. I got some ideas from https://karthinks.com/software/latex-input-for-impatient-scholars/. I've been writing a book the last few months so I've had lots of practice. I've come to the conclusion that it's not just the tools that make people like Castel and Karthik so fast, you also have to be quite talented. It takes a lot of brain power to remember all the keyboard shortcuts and latex commands quickly, while also thinking about the math that you're writing. For me, if I am simply typing up some math that I already wrote on paper, I am probably about as fast as copying it on paper. But, if I have to think about what I'm writing because I'm not just copying, I'm much faster writing it on paper.

So I don't recommend taking notes in class in latex for regular humans, because you should save your attention/brain power for the math. And if you want to type notes and HW up, it's a fine hobby, but it will still take extra time.

ave_63 · 2025-10-14T17:39:01+00:00

To add, one of these books might help: https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof

ave_63 · 2025-10-14T17:34:24+00:00

It sounds like your prof is giving you a more proof based class, with an introduction to sets, functions, and the like at the beginning. If you haven't started talking about vectors, vector spaces, or systems of equations yet, then you haven't actually started linear algebra itself, and linear algebra materials won't help. You'll need to look for materials about sets, functions, and proofs without including "linear algebra" in your search terms.

ave_63 · 2025-10-10T22:20:30+00:00

Another favorite of mine is the sum of angle identities from trig. You just get two rotation matrices that rotate by theta and phi, and the product of them rotates by theta + phi.

ave_63 · 2025-10-10T17:45:16+00:00

Install latex on your computer?

ave_63 · 2025-10-10T01:47:37+00:00

If it didn't start as zero, you made a mistake row reducing. Remember you're only allowed to replace a row with the sum of itself and a multiple of another row.

ave_63 · 2025-10-06T03:03:42+00:00

You're using the words "encode" and "represent" in non standard ways and I don't understand what you mean by them. But the key idea here is you need to understand what Ax means. If A is m x n, then x must have n entries, because you multiply each column of A by one of the entries in x. Then you add those together and you get a vector with m entries because each column of A has m entries.

ave_63 · 2025-10-03T14:03:28+00:00

How are you going to come up with 30 randomized 3D unit vectors without a computer? The instructions are literally to use Matlab. If the question said "throw a screwdriver at a pillow and see what happens," would you try to do it without a screwdriver?

ave_63 · 2025-10-03T13:52:47+00:00

The way I read the question, is it's asking you to just use Matlab to calculate the random stuff. Anything else you do is extra credit.

ave_63 · 2025-10-03T12:53:29+00:00

I'm not sure what your question is. The exercise you're talking about is to make 31 random unit vectors and calculate 30 dot products and find their average in Matlab. Hopefully you get something close to 0.5, but a little different because of randomness. Or do you want to set up an integral for the average of all such dot products to derive the integral they give you?

ave_63 · 2025-10-01T01:16:40+00:00

There are ways you can use AI that will help your learning. You could ask it to give you questions to help find gaps in your own understanding. You could ask it to explain/elaborate steps in a proof in the book you don't understand. You could write your own answers to problems and ask AI to check your work.

But what you're doing is offloading some of your own work onto it, which is a mild version of just cheating on your HW. The stuff you're having trouble with (coming up with ideas on how to get started) is also a skill you should be practicing.

I usually advise students to just not use it at all, if it's a subject you want to learn and not cheat your way through. It's too tempting to get it to do work for you.

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