https://8gwifi.org/physics/labs/circuit-simulator.jsp

This material focuses not on pure linear algebra, but rather on its applications in engineering and quantum mechanics.
Geometrically, the meaning of an eigenvalue signifies the scaling (expansion or contraction) and occasionally the inversion of an invariant coordinate axis.

Hi

If you are remotely interested in understanding what bits of linear algebra are used in defining the Gate model framework Quantum Computing, oh boy this is for you. I am the Dev behind Quantum Odyssey (AMA! I love taking qs) - worked on it for about 10 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 12yo+ 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 & learn a ton about

• 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.

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

Hi guys,

Super excited about this update to Linear Algebra Visualiser - which now includes matrix composition, the ability to add a translation matrix and an expanded Step by Step explanation.

Have a look at the video above for a detailed demo!

PS: These new features are available as In App Purchases but you get 1 week as a free trail so you can always check it out and cancel if you are not happy.

PS 2: For people who were Beta Testers - I need some more time to setup the ability to give out offer codes (it’s all complicated with Apple).

Thank you all for your support, it means a lot!

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

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

Web Demo: https://sockerjam.github.io/LinearAlgebraVisualizerWeb/

Last three images of this post are GraphMath's work. Here is the link to their original post: https://www.reddit.com/r/LinearAlgebra/s/cdzAjKVtOc

My writing focuses on what eigenvectors are really telling us, why complex eigenvectors matter, and how one complex eigenvector can encode a two-dimensional, real invariant plane.

Hi everyone!

I wanted to share this quick thought that has really helped me with linear algebra;

Let's say: P = (1, 0)

This vector/point is simply the result of scaling the basis vectors ihat [1, 0] and jhat [0, 1]:

Now, if we apply a transformation to the basis vectors, let's say a 90 degree rotation, the same process applies. For example, after the transformations each basis vector lands at column, ihat [0, 1], jhat [-1, 0].

Scaling this with our vector, [1, 0] gives us [0, 1].

So, a transformation transforms the basis vectors and our vector scales it, that's it :)

Enjoy!

manic is a tiny language for making animations. You write a short text file; manic renders a smooth, glowing video. No timeline scrubbing, no keyframes by hand — you describe what’s on screen and when things happen, and the engine does the rest, deterministically.

• Try manic
• manic docs

Manic Animation code

// ============================================================================
//  Essence of Linear Algebra, chapters 1→2→3 in one arc.
//
//  Ch1  A vector is an arrow from the origin; its coordinates are [x, y].
//  Ch2  Those coordinates are SCALARS: v = x·î + y·ĵ — a linear combination of
//       the basis vectors î=(1,0), ĵ=(0,1).
//  Ch3  A matrix is a transformation: it says where î and ĵ LAND, and the whole
//       grid morphs to follow (`gridmap` animates it). Because v = x·î + y·ĵ,
//       it rides along, landing at x·(new î) + y·(new ĵ).
//
//  Example: v = [1, 2] under [[2,1],[1,2]] (î→(2,1), ĵ→(1,2)), so v lands at
//  1·(2,1) + 2·(1,2) = (4,5). Origin (400,470), 55 px/unit.
// ============================================================================
title("Vectors, basis, and transformations");
canvas(1280, 720);
template("black");

text(narr, (640, 696), "A vector is an arrow from the origin.", 24);

// the plane + basis, ready to be transformed in Ch3 (starts at the identity)
gridmap(gm, (400, 470), 55, 2, 1, 1, 2, 5);

// the vector v = [1,2] and its linear-combination scaffold
vector(v, (400, 470), (55, 110), yellow);
vector(jj, (455, 470), (0, 110), red);           // 2·j-hat, stacked on i-hat → reaches v
equation(vlab, (486, 352), `\vec{v}=[1,2]`, 30); color(vlab, yellow);
equation(ilab, (472, 500), `\hat{\imath}`, 26); color(ilab, lime);
equation(jlab, (366, 414), `\hat{\jmath}`, 26); color(jlab, red);
equation(mtx, (980, 165), `\begin{bmatrix}2&1\\1&2\end{bmatrix}`, 44);

// Ch4 — a second transform (90° rotation) composed on top: morph M1 → M2·M1
gridmap(gm2, (400, 470), 55, -1, -2, 2, 1, 5, 2, 1, 1, 2);
equation(comp4, (640, 96), `\begin{bmatrix}0&-1\\1&0\end{bmatrix}\begin{bmatrix}2&1\\1&2\end{bmatrix}=\begin{bmatrix}-1&-2\\2&1\end{bmatrix}`, 34);

// the determinant cell (Ch5), the projection (Ch7), and the alt-basis grid (Ch8)
determinant(dt, (400, 470), 55, 2, 1, 1, 2, gold);
project(pr, (400, 470), 55, (1, 2), (3, 1));       // project v=(1,2) onto span(w=(3,1))
squish(sq, (400, 470), 55, 1, 2, 5);               // Ch7 duality: dot with v = squish onto a line
eigen(ev, (400, 470), 55, 2, 1, 1, 2, gold);       // Ch9 eigenvectors of [[2,1],[1,2]] (λ = 3, 1)
diagonalise(dg, (400, 470), 55, 2, 1, 1, 2, gold); // Ch10 eigenbasis = a pure stretch
coords(cf, (700, 450), (-4, 4), (-3, 3), 46, 46, 0); // Ch11 a function graph…
plot(fn, (700, 450), 46, 46, "0.25*x*x*x - x", (-3.6, 3.6)); color(fn, mint);
equation(deq, (700, 130), `\tfrac{d}{dx}\;\leftrightarrow\;\begin{bmatrix}0&1&0\\0&0&2\\0&0&0\end{bmatrix}`, 32);
gridmap(cb, (400, 470), 55, 1, 0.7, 0.4, 1, 5);
vector(vb, (400, 470), (55, 110), yellow);

// --- initial visibility ---
hidden(gm.i); hidden(gm.j);                       // basis revealed in Ch2
hidden(v); hidden(vlab); hidden(jj); hidden(ilab); hidden(jlab); hidden(mtx);
hidden(dt); hidden(dt.unit); hidden(dt.val); hidden(pr);
hidden(sq); hidden(sq.line); hidden(sq.dual);
hidden(ev); hidden(dg); hidden(cf); hidden(fn); hidden(deq);
hidden(gm2); hidden(gm2.bg); hidden(gm2.i); hidden(gm2.j); hidden(comp4);
hidden(cb); hidden(cb.bg); hidden(cb.i); hidden(cb.j); hidden(vb);

// --- Ch1: a vector ---
wait(0.5);
par { show(v); show(vlab); }
wait(1.3);

// --- Ch2: coordinates are a linear combination of the basis ---
say(narr, "Its coordinates are scalars: v = 1·i-hat + 2·j-hat.");
par { show(gm.i); show(gm.j); show(ilab); show(jlab); }
wait(0.7);
show(jj);                                         // i-hat then two j-hats reach v
wait(1.6);

// --- Ch3: the matrix transforms space; v rides along ---
say(narr, "A matrix moves i-hat and j-hat — and the whole grid follows.");
show(mtx);
par { fade(jj); fade(ilab); fade(jlab); fade(vlab); }
par { to(gm, morph, 1, 2.2); grow(v, (620, 195), 2.2); }   // space deforms, v follows
wait(0.6);
say(narr, "v lands at 1·(new i-hat) + 2·(new j-hat) = (4,5).");
wait(1.6);

// --- Ch4: matrix multiplication is composition ---
say(narr, "Ch 4 — apply one transform, then another: that is matrix multiplication.");
show(comp4);                                       // the product, clearly placed up top
par { fade(gm); fade(v); }
show(gm2);                                          // starts at M1 (seamless with Ch3)
to(gm2, morph, 1, 1.8);                             // now rotate 90°: the grid is at M2·M1
wait(1.3);
to(gm2, morph, 0, 1.5);                             // undo the second transform, back to M1
par { fade(gm2); fade(comp4); }
par { show(gm); show(v); }
wait(0.6);

// --- Ch5: the determinant is the area scale factor ---
say(narr, "Ch 5 — the determinant is how much areas scale: 2·2 − 1·1 = 3.");
par { show(dt); show(dt.unit); show(dt.val); }
wait(1.9);
par { fade(dt); fade(dt.unit); fade(dt.val); }    // fade = timeline hide (hidden is base-state only)

// --- Ch6: column space + the inverse undoes the transform ---
say(narr, "Ch 6 — the columns span the whole plane, so the inverse sends space back.");
par { to(gm, morph, 0, 1.8); grow(v, (455, 360), 1.8); }   // reverse the morph = the inverse
wait(1.4);

// --- Ch7: the dot product — projection, and its dual (a squish onto a line) ---
say(narr, "Ch 7 — the dot product of v and w is a projection — w's shadow on v.");
par { fade(gm); fade(gm.bg); fade(v); fade(mtx); }
show(pr);
wait(1.6);
say(narr, "It's also a squish of the whole plane onto a line — the dual vector IS v.");
fade(pr);
par { show(sq); show(sq.line); show(sq.dual); }
to(sq, morph, 1, 1.9);                              // collapse space onto the number line
wait(1.0);
to(sq, morph, 0, 1.2);                              // and back — the dual vector remains
wait(0.8);

// --- Ch8: change of basis ---
say(narr, "Ch 8 — change of basis: the same arrow, read on a different grid.");
par { fade(sq); fade(sq.line); fade(sq.dual); }
par { show(cb); show(vb); }
to(cb, morph, 1, 1.8);                                       // the coordinate grid changes; v stays put
wait(1.8);

// --- Ch9: eigenvectors and eigenvalues ---
say(narr, "Ch 9 — some vectors keep to their own line: eigenvectors (λ = 3 and λ = 1).");
par { fade(cb); fade(vb); }
show(ev);
wait(2.0);

// --- Ch10: the eigenbasis is a pure stretch ---
say(narr, "Ch 10 — in the eigenbasis, the transformation is just a diagonal stretch.");
fade(ev);
show(dg);
wait(2.0);

// --- Ch11: abstract vector spaces ---
say(narr, "Ch 11 — even functions are vectors; the derivative is a linear map with a matrix.");
fade(dg);
par { show(cf); show(fn); show(deq); }
wait(2.2);

This is a follow-up to our previous post on how the choice of free variable changes the real and imaginary parts of a complex eigenvector:

https://www.reddit.com/r/LinearAlgebra/comments/1vl5s28/complex_eigenvectors_of_a_22_matrix_how_the_free/

For a general 2×2 real matrix with complex eigenvalues, changing the phase of the free variable rotates and scales the pair Re(x̃₁), Im(x̃₁) through a matrix-transformation ellipse.

Here we look at the special case where the original matrix itself is a rotation-scaling matrix.

In this case, for every nonzero choice of the free variable, Re(x̃₁) and Im(x̃₁) remain perpendicular and equal in length. The numerical examples at the bottom show the same phase experiment as in the previous post, but now the ellipse becomes a circle.

This is the next page of the chapter on complex eigenvectors and the rotation-scaling theorem. More to follow.

This is a follow-up to our previous post on how the choice of free variable changes the real and imaginary parts of a complex eigenvector:

https://www.reddit.com/r/LinearAlgebra/comments/1vl5s28/complex_eigenvectors_of_a_22_matrix_how_the_free/

For a general 2×2 real matrix with complex eigenvalues, changing the phase of the free variable rotates and scales the pair Re(x̃₁), Im(x̃₁) through a matrix-transformation ellipse.

Here we look at the special case where the matrix itself is a rotation-scaling matrix.

In this case, for every nonzero choice of the free variable, Re(x̃₁) and Im(x̃₁) remain perpendicular and equal in length. The numerical examples show the same phase experiment as in the previous post, but now the ellipse becomes a circle.

Thanks to u/StanleyDodds for pointing out an unnecessary detour in the previous version. We have corrected and simplified it here.

This is the next page of the chapter on complex eigenvectors and the rotation-scaling theorem. More to follow.

This material is not the pure linear algebra or functional analysis covered in mathematics departments. It is applied material utilized in engineering and quantum mechanics.
We will also cover the inverse Fourier transform, which localizes position by extending a finite-dimensional vector space to an infinite-dimensional vector space and summing an infinite number of momentum bases.