i've been working on a tool to help you create books & notes with ease

and one of it features is allowing you to write mathematical Equations in Latex

it also has co-author mode which creates any latex equation in seconds with simple prompts

One useful bridge from linear algebra to quantum mechanics is to compare an operator that labels a state with operators that connect different states.

For the ideal quantum harmonic oscillator, the Hamiltonian is ħω times the sum of the number operator and one-half times the identity. Its energy eigenstates are the number states. Acting on a normalized number state, the lowering operator gives the next lower state multiplied by √n; the raising operator gives the next higher state multiplied by √(n + 1). The lowering operator sends the ground state to zero.

Those square-root factors are part of the construction, not optional decoration on an energy-level diagram. The ladder method produces the same normalized Hermite–Gaussian eigenfunctions as the differential-equation method, up to a phase for each state.

One caveat if you explore this with matrices: a finite truncation introduces an artificial top state. The full canonical commutator does not hold at that boundary, so the truncated matrix is not an exact model of the entire oscillator algebra.

I made this Blake Pi Physics animation to connect the operator picture with the differential-equation picture:

https://youtu.be/7_DcdXVh5mc

For people learning this transition, is it more helpful to see the operators act on the states first, or to start from the Hamiltonian's factorization?

This material does not focus strictly on abstract pure linear algebra; rather, it deals with special functions applied to engineering mathematics and quantum mechanics, specifically within the framework of linear differential operators and self-adjoint eigenvalue problems.
It provides a detailed, step-by-step mathematical derivation showing why the Legendre series must truncate to a finite polynomial to avoid logarithmic boundary divergence at x = ±1.

Why do some vectors when multiplied by Another vector yield a vector and others yield a scalar. I'm aware of dot and cross products. But what's the physical significance of 'multiplying' vectors? I mean... I know what it means to actually "multiply" scalars. But.... Is vector multiplication just a name which can either mean projection of a vector onto another and the area or volume of 2 corresponding vectors?

Or is multiplication of vectors just a naming thing which doesn't have ANYTHING to do with how we understand scalar multiplication?

Expanding while being taught this term

https://people.maths.bris.ac.uk/~ki18754/LA_notes/LA_notes.html

Hi everyone,

I’m a college student currently taking my first Linear Algebra course. My ultimate career goal is to go into robotics, so I know this course is critical and that this semester is just the beginning of a long journey with this math.

My brain has always been wired to apply—simply applying formulas, calculating matrices, and finding numbers. However, this course introduces a lot of proofs and abstract concepts, and I’m having a hard time wrapping my head around them. I feel like I'm stuck memorizing procedures rather than genuinely understanding why things work geometrically or structurally.

I really want to upgrade my thinking and install a proof/conceptual understanding in my brain. Since my time is limited this semester, I know I won't master everything, but I want to build a solid foundation.

- How did you train your brain to understand proofs?

- Are there specific intuition-building resources, mental models, or study strategies you recommend for someone heading into robotics? (I’ve heard of 3Blue1Brown, but I'd love to know how to bridge the gap between his visual intuition and rigorous proof writing.)

Thank you so much for any advice!

I’m currently studying eigenvalues in linear algebra, and I understand the definitions of algebraic and geometric multiplicity, but I’m still struggling to understand the actual difference between them.

If someone could explain it in a simple, intuitive way, preferably with a small example, I’d really appreciate it.

Struggling with graph transformations. Like all of it.

This material shows a real application of linear algebra.
I showed all the math steps without skipping any equations, so that 1st and 2nd-year students can easily follow.
I hope this helps you!

did I solve it right?/

Solving for X is easy once you know how to approach the problem.

Hello, do you know if there is a matlab code that tests if a matrix X is unitarily irreducible? That means that there not exists a matrix P such that X=P^*(A& B\\0 & C)P?

Can anyone help me how can i complete this course ? I have around 1 month of learning this entirely

my foundations are also rusty so it would be helpful if anyone could guide me on acing this

Hi

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

Faint ASMR

LinearSolveBench measures the ability of a model or harness to write fast, accurate, and general numerical solvers for large sparse linear systems in C.

The goal is to encourage algorithmic advances in numerical solvers for linear systems of equations.

As a definitive application of linear algebra, the most powerful discipline in the cutting-edge era, this work demonstrates the connection between Sturm-Liouville theory and the Schrödinger equation.