Hello everyone,

You might have seen my previous posts about the Linear Algebra Visualizer app (the app that helps with visualising linear transformations).

You guys have already been amazing with your support which is why I'd love to invite you as a beta tester.

I'm currently working on expanding the app to include affine transformations and matrix compositions but I really want to get it right and for it to continue to be a great learning tool.

This will be specific to the Apple platforms - iOS, iPad and MacOS - and all you need is the TestFlight app by Apple and fill in your details in the form below: https://forms.gle/98dHacjaAeszhNvr6

(As a Beta tester you will also get the future premium content for free)

Thank you and I look forward to your feedback!

We made an animated visualization of QR decomposition using Householder reflections.

This uses the same matrix as our earlier Gram–Schmidt and Givens rotation QR animations, so the three methods can be compared on the same example.

Householder QR reaches an upper triangular matrix by reflecting the active part of the matrix column by column. In this 3×3 example it takes two reflections, compared with three Givens rotations, which eliminate one entry at a time.

A related page with another Householder QR example is here:
https://www.graphmath.com/la/visuals/householder-reflection.html

We welcome feedback on clarity, pacing and presentation.

Hi
Excited to be able to announce that QO is almost ready to leave Early Access! This month I published a large patch that covers more than a year of work (lots of analytics, I've been tracking where ppl were getting stuck). Thank you a ton for your support, this game has seen a lot of love from this community. Game is almost done.

If you are interested in a highly intuitive visual method that faithfully describes all universal quantum computing and physics behind, this is for you. I am the Dev behind Quantum Odyssey (AMA! I love taking qs) - worked on it for about 10 years (3.5 in phd), 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 (that was actually my PhD research) 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 covered

• 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.
• Build & See Quantum Algorithms in Action – instead of just writing/ reading equations, make & watch algorithms unfold step by step so they become clear, visual, and unforgettable. Quantum Odyssey is built to grow into a full universal quantum computing learning platform. If a universal quantum computer can do it, we aim to bring it into the game, so your quantum journey never ends.

Streams to watch:

khan academy style tutorials on qm/qc: https://www.youtube.com/@MackAttackx

Physics teacher wholesome stream with over 500hs in https://www.twitch.tv/beardhero

I am fully confident that this detailed algebraic approach will be of great help, especially to students in applied engineering fields.

While this derivation is fundamentally rooted in quantum mechanics, the mathematical structure is essential for modern engineering.

Specifically, if you are majoring in Electronic and Semiconductor Engineering, Quantum Engineering, or Materials Science, you will find this material highly practical.

Understanding these matrix mechanics and operator algebras is the absolute foundation for dealing with qubits in quantum computing, as well as spin-interactions in modern spintronics and advanced device physics.

The topics are challenging, but this is presented more accessibly and in greater detail than any other textbook.

I hope this supports your practical engineering applications. by Taeryeon.

Matrix Calculator

13 operations on one page. Determinant, inverse, eigenvalues, multiplication, and more — with step-by-step solutions.

https://8gwifi.org/math/matrix-calculator.jsp

If I am asked to find 2 unit vectors that are orthogonal to a single vector v (5,-2). Here is what I did:

1- find 2 vectors orthogonal to v, both of them are basically the same because if I multiply the first vector I found by some scalar, I get the second vector.

2- Then I normalized each vector that I got in 1. Here comes the problem, when I normalize the vectors, I get the same answer for both vectors because the two vector are essentially the same as explained in 1). The questions mentions that I need to find two unit vectors and I end up with only one.

We made an animated visualization of QR decomposition using Givens rotations.

This uses the same matrix as our earlier Gram–Schmidt QR animation, so the two methods can be compared directly.

The animation shows how successive rotations zero out entries below the diagonal while preserving lengths and angles. In 3D, each Givens rotation is a rotation in one coordinate plane, so the columns visibly move until the matrix becomes upper triangular:

A = QR

This is the same QR goal as Gram–Schmidt, but the geometric action is different: instead of projecting and subtracting components, Givens rotations turn vectors in coordinate planes until selected components disappear.

For this animation and a related Givens rotation visualization, see:
https://www.graphmath.com/la/visuals/givens-rotation-algorithm.html

We welcome feedback on clarity, pacing and presentation.

I'm taking Linear Algebra for the first time and I'm not sure how to solve this problem. Could anyone kindly explain how they determined the correct answers?

for 2 by 2 I understand it might that

[1 0; 0 1] technically isn't Lower but [2 3; 0 1] is definitely an upper for an answer of the same

but for 3x3 is it right that this answer is correct and thus achieving A?

[1 0 0; 1 1 0; 1 0 1] = L

[ 1 1 0; 0 1 1; 0 0 2] = L

for A = LU

Thanks, or maybe i misunderstood the question

Hi everyone, I am looking for a free PDF for this book but not the global edition. Thanks

I’m self learning linear algebra and I was wondering if Symbolab is an accurate ai. What is the most accurate ai to help me learn linear algebra?

Hello everyone,

Like what my post suggests, I'm looking on how to pass my linear algebra class. It is a lower division class and for some reason these topics just don't "click" for me. My class uses Linear Algebra and Its Applications by Lay et al., 6th edition. I've added the course topics I learn this quarter.

My current grade is a 54.88%, and is composed of 2 midterm exams (making up 30% of my entire grade). Midterm 1 I did okay-ish at a 31/40, but bombed midterm 2 at 22/40. I am also graded via 5 quizzes which make up 25% of my total grade. Quiz 1 I got 7/10, and quizzes 2/3 I got 8/10. I took quiz 4 and didnt think I did too well. I take quiz 5 in 2 weeks. Participation is 5%, so its essentially free points. I also have homework which is 10%. I do fine on the homeworks (and also the quizzes and discussion worksheets), but for some reason I just don't do good on the exams. The lowest quiz also gets dropped.

I had this specific professor last quarter for Differential Equations and got an A- in that class. He has a policy that if the final is higher than the lower of the 2 midterms (for me itll be midterm 2), then the final exam will replace that midterm grade. I asked Claude, and it said I'll need around a 75% to get a 80%/B- overall. I don't think getting an A (or A-) is really in my reach.

As of now, I've been reviewing my book, doing the worksheets + quizzes, and reviewing lecture notes. Any other ideas on how to get this 75% for the final? Please help!!

Has Anyone tried this book? I've been following his NPTEL lectures and it's been intuitive.

We made an interactive visualization of Gram–Schmidt as QR decomposition:
Gram Schmidt Stepwise Visualization

It shows each column being projected, corrected, normalized, and added to Q, while the corresponding coefficients form R. The goal is to make the geometry behind the usual algebraic steps visible: projections, orthogonal components, scaling, and the final A = QR.

We welcome feedback on presentation and clarity

This post does not present rigorous, pure linear algebra that demands strict mathematical proofs. Instead, it focuses on the application of linear algebra in fields such as quantum mechanics, control systems, and quantum computing.

 By providing concrete examples within these applied disciplines, this text demonstrates how linear algebraic concepts are practically utilized and illustrates why the study of linear algebra is essential.

 In particular, it will clearly and easily assist the learning of those studying fields that require an algebraic approach to angular momentum. By Taeryeon.

I made a visualization of cofactor expansion for a 3×3 determinant.

The idea is to start with the usual interpretation of det(A) as signed volume, then show each cofactor term as:

• one coordinate projection of the expansion column

• one projected 2×2 minor in the orthogonal coordinate plane

So instead of treating

det(A) = a₁₁M₁₁ − a₂₁M₂₁ + a₃₁M₃₁

as only a symbolic rule, the goal is to make the three terms look like signed volume pieces.

The image shows the three terms for one example matrix, including the “same sign / negated sign” behavior of the cofactor signs.

Full explanation, including determinant sign:

https://www.graphmath.com/la/determinant/determinant-sign.html

I would be interested in feedback on whether the geometric interpretation is clear

I felt so confident walking into it, I was sure I could get an 80. I got it back and saw I got in the mid-30s.

Even worse, I knew what was happening, and all of the answers I had done before since they were homework questions.

I haven’t gone to office hours yet, planning to, but I’m just so lost. I know my answers were correct, but I’m not sure why I would do so poorly.
It’s a partially proofs based course, so I understand I probably overcomplicated my answers or didn’t answer how they wanted me to, but damn

Anyone got this textbook?! :)) Its for Math 1600A would hella appreciate anyone who got it please and thank you.

Hi guys,

Just wanted to say thank you for all the positive feedback from my first post about Linear Algebra Visualizer.

Especially from u/LongjumpingEar7568 and u/Ron-Erez who suggested adding the eigenvectors and values. This is now complete and I’m excited to share it with you guys in this community.

This update includes:

- Explore eigenvectors and eigenvalues visually

- Polygon presets

- Enter any 2x2 matrix

Thanks for your supports and let me know of any feedback in the comments please :)

Available on the platforms below:

Mac: https://apps.apple.com/gb/app/linear-algebra-visualizer/id6763524968

Linear Algebra Visualizer

iOS/iPad: https://apps.apple.com/us/app/linear-algebra-visualizer/id6763524968

Web: https://sockerjam.github.io/LinearAlgebraVisualizerWeb/ (Eigenvectors and values coming soon!)

This content is not intended for rigorous, proof-oriented linear algebra as practiced in mathematics departments.

Rather, it focuses on applied linear algebra suitable for immediate implementation in engineering and applied sciences.

The discussion primarily addresses the derivation of inverse matrices through Gauss-Jordan elimination and the adjugate matrix method.

I hope you find this useful for your work.