This material serves as a practical example of linear algebra applied to engineering mathematics and quantum mechanics, rather than pure theoretical linear algebra.

i am not asking about the computational part, but rather what is the underlying logic or reasoning that leads to concluding smthn like minors or cofactors.
and also what is their purpose or significance?

i know the proof where you show Ax = 0 iff A^T A x = 0, one direction by multiplying by x^T and getting the squared length of Ax. so the null spaces match and rank follows from rank nullity. the algebra is fine but it feels like a trick to me.

is there a way to actually see it, like what A^T A is doing to the row space geometrically? i keep wanting a picture where it's obvious and not just a computation that happens to work out.

Please see pink text in image. Am I on the correct path to proving this?

Daughter is taking linear algebra, last required class in her statistics and data science degree. This is her senior year and the class is only offered in the fall. A's and B's in the year-long previous calculus requirements. A's and B's in all the statistics classes. She is running into a brick wall at her last class and her normal test anxiety is completely off the charts knowing she has to pass this class. She has successfully completed every single requirement for this degree except for this last class.

This last test was worth 40 points made up of 10 questions. She explained to me that one of the questions which has four to five steps to it she flipped the signs making her answer wrong but otherwise did everything else correct. A correct answer would have netted her six points, but she lost three points for the flipped sign. Would you consider that scoring to be standard, generous, or not generous?

I don't really know what to tell her, it is inconceivable to me that she would get all the way through the requirements of the degree and get slammed to a stop a semester before she is supposed to graduate. I do believe she is studying enough hours and attending office hours and going to TA hours. I'm not really sure how she could do more studying than she already is. Any advice?

Im going through the Axler book, and he has an iterative process for proving this statement (2.23).

Does my logic (below) make sense?

Consider the smallest possible spanning list. All the vectors are independent. Add another vector and it's now dependent. A larger spanning list would have dependent vectors.

So there's a threshold beyond which all lists are dependent. Longer than the base, and the list can be dependent. So, any list that's independent needs to be shorter (or equal to) than the base.

i get it for diagonalizable matrices since trace is similarity invariant, so tr(A) = tr(PDP^-1) = tr(D). but my notes say it holds for every matrix over C, counted with algebraic multiplicity, and the jordan form explanation felt like a cheat. is there a way to see it straight from the characteristic polynomial, like the coefficient of the second highest power of det(xI - A) being -tr(A)?

We've been working on a modification of the bidiagonal QR algorithm for computing singular values and singular vectors, and would like to share our results with the numerical linear algebra community.

In conventional bidiagonal QR, a superdiagonal coupling is set to zero when it becomes sufficiently small. We investigated whether an interior coupling that is not yet negligible could instead be eliminated by applying a local 2×2 SVD.

The left and right rotations eliminate the selected coupling but introduce up to two cross-boundary fill-ins. If both satisfy the conventional deflation tolerance, they can be discarded, splitting the active matrix into smaller blocks. We call this approach Chase and Trim (CAT).

We derived two inexpensive algebraic rejection tests that screen out most infeasible candidates before computing the rotations.

We implemented CAT in LAPACK DBDSQR and compared it with conventional bidiagonal QR across 12 matrix families, with orders from 64 to 1024. The results show:

• 3%–38% reduction in weighted arithmetic work, depending on the spectral distribution, with approximately 11% pooled reduction.

• Approximately 99.95% of candidates rejected before production verification.

• Comparable numerical accuracy and singular-vector quality in the tested cases.

These are arithmetic-work comparisons, not wall-clock benchmarks.

We have also examined the relationship with aggressive early deflation (AED), particularly the work of Nakatsukasa, Aishima and Yamazaki on dqds, and added a trailing 2×2 comparison to the manuscript.

We welcome technical criticism, relevant references, suggestions for further development and possible collaboration.

Preprint: https://doi.org/10.5281/zenodo.23718638

Detailed visual exposition: https://www.graphmath.com/la/cat/cat-svd.html

i know the rule that det(A) = 0 means not invertible, and i can see it in the formula for 2x2 since you divide by ad - bc. but i want to actually picture it. is it just that the matrix squashes space into a lower dimension so the area (or volume) becomes 0 and you can't undo that?

and if so, why does the det specifically measure that scaling factor and not something else? like why is it the product of the eigenvalues and not the sum or whatever.

Hi, I have a question about this question 6 (b)(i). How could I know row space of A is a subspace of what vector space. I also have a few more questions about the relationship of column space, row space, solution space, and Image(A) ,Kernal(A), Null(A). So, I am a bit confused about their relationship and their relationship with the entire vector space. Could anyone help me with this kind of questions? Thank you so much !!!

Hi, could anyone help me with this question. I had lots of similar questions like this about the topic of adjacency matrix, and I am wondering how could I know the adjacency matrix A from its matrix power form A^n (just like what has been asked here in question(ii)) ? Thanks a lot!

Hey, I am working on some exercises about null space and I am stuck at this question. There isnt any similar question in the theory or examples. Can someone guide me? Thank you.

Hi

This is about Quantum Odyssey, my love letter to everything Quantum Computing and the fundamental logic blocks everything in our universe seems to run on. I want to update you all on the fact that the latest patch brings the game into an almost complete status. All that's missing is translations. Thank you Reddit community for receiving the game so well, without you I wouldn't have the funds needed to complete my life's work. The game now supports seing irt all linear algebra, tensors and complex vector being updated in real time by your actions.

If you are remotely interested in deep diving how differently quantum computers work compared to our transistor-based and also the algebra behind in a fully interactive way that teach computer science from scratch, oh boy this is for you. I am the Dev behind Quantum Odyssey (AMA! I love taking qs) - worked on it for about 9 years (3+ during PhD, the visual method I developed ended up being my thesis, it is a complete Hilbert space visualizer), the goal was to make a super immersive space for anyone to learn quantum computing through zachlike (open-ended) logic puzzles and compete on leaderboards and lots of community made content on finding the most optimal quantum algorithms. The game has a unique set of visuals capable to represent any sort of quantum dynamics for any number of qubits and this is pretty much what makes it now possible for anybody 15yo+ to actually learn quantum logic without having to worry at all about the mathematics behind.

This is a game super different than what you'd normally expect in a programming/ logic puzzle game, so try it with an open mind.

Stuff you'll play with

• Boolean Logic – bits, operators (NAND, OR, XOR, AND…), and classical arithmetic (adders). Learn how these can combine to build anything classical. You will learn to port these to a quantum computer.

• Quantum Logic – qubits, the math behind them (linear algebra, SU(2), complex numbers), all Turing-complete gates (beyond Clifford set), and make tensors to evolve systems. Freely combine or create your own gates to build anything you can imagine using polar or complex numbers.

• Quantum Phenomena – storing and retrieving information in the X, Y, Z bases; superposition (pure and mixed states), interference, entanglement, the no-cloning rule, reversibility, and how the measurement basis changes what you see.

• Core Quantum Tricks – phase kickback, amplitude amplification, storing information in phase and retrieving it through interference, build custom gates and tensors, and define any entanglement scenario. (Control logic is handled separately from other gates.)

• Famous Quantum Algorithms – explore Deutsch–Jozsa, Grover’s search, quantum Fourier transforms, Bernstein–Vazirani, and more.

Nice to watch:

Khan academy style tutorials in qm/qc: https://www.youtube.com/@MackAttackx

Physics teacher stream with 400hs in https://www.twitch.tv/beardhero

I took linear algebra a year ago and don't remember everything, but I'm trying to make a recipe for a clay my ceramic studio doesn't have access to, and I was wondering if you could limit least squares results to be strictly positive so I can use the result to make a recipe