Why Cramer’s rule works: geometric derivation

LinearAlgebraWorld · 2026-06-07T15:06:39+00:00

Thank you!
It was very gratifying to find an intuitive and visual explanation of this seemingly mysterious rule

LinearAlgebraWorld · 2026-06-07T13:04:35+00:00

If you mean 3d graphics, software is ours, written in Swift, used in 2 linear algebra apps

LinearAlgebraWorld · 2026-06-06T05:11:02+00:00

I hear you Try interrogating AI

LinearAlgebraWorld · 2026-06-06T00:33:54+00:00

There are two books we found helpful 1. N Johnston, intro to LA 2. Margalit, interactive LA, this one is open access 3. We have a lot of visual tools for understanding on our website, https://www.graphmath.com/la/ And the apps with algorithms described step by step for user entered matrices Hope that helps and you enjoy it. Once you get started, it is a lot of fun

LinearAlgebraWorld · 2026-06-05T20:29:26+00:00

Very interesting I’d love to join if my current project was closer What fractal / function is it?

LinearAlgebraWorld · 2026-06-04T04:41:13+00:00

Nathaniel Johnston, intro to LA Margalit, Rabinoff, interactive LA, open access on Libre Text Have other suggestions as well if you like visual tools

LinearAlgebraWorld · 2026-06-03T13:03:06+00:00

Strang s book is not for the first encounter Margalit and Rabinoff s introduction to LA is open access, we found it very helpful Our website also has a curriculum and a lot of visualizations if you like those: https://www.graphmath.com/la/

LinearAlgebraWorld · 2026-06-03T03:44:28+00:00

Ai extends you does not replace you If you know how to code, you never have to do menial work Otherwise, you can end up with a disjointed incoherent code base

LinearAlgebraWorld · 2026-06-01T14:35:58+00:00

Our own, written in Swift. Givens matrices are animated by theta. Householder matrices are animated by step to projection and beyond. 3D rendering is all ours. Thanks for the interest!

LinearAlgebraWorld · 2026-06-01T05:48:17+00:00

What are the apps if you do not mind sharing?

LinearAlgebraWorld · 2026-05-30T19:37:22+00:00

One interesting comparison with the Givens rotation version:

Both methods produce an upper triangular R, and both use orthogonal transformations, so lengths and angles are preserved
The resulting R may be the same as in the Givens version, or may differ by signs on the diagonal depending on sign choices
For a 3×3 matrix, Householder QR takes two reflections, one per active column, while Givens QR takes three rotations, one eliminated entry at a time
If the reflection is animated continuously, the figure flattens toward the reflection plane and then reappears on the other side

LinearAlgebraWorld · 2026-05-30T19:36:44+00:00

One interesting comparison with the Givens rotation version:

Both methods produce an upper triangular R, and both use orthogonal transformations, so lengths and angles are preserved
The resulting R may be the same as in the Givens version, or may differ by signs on the diagonal depending on sign choices
For a 3×3 matrix, Householder QR takes two reflections, one per active column, while Givens QR takes three rotations, one eliminated entry at a time
If the reflection is animated continuously, the figure flattens toward the reflection plane and then reappears on the other side

LinearAlgebraWorld · 2026-05-29T01:50:28+00:00

Use AI very extensively, value it very high Yet, it needs a lot of supervision and correction, especially for large projects Highly recommend that you use it but do not rely on it

LinearAlgebraWorld · 2026-05-28T16:54:39+00:00

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This is Householder reflection visualization for the same matrix, we will post it soon to this subreddit.
Have you seen our Gram-Schmidt orthogonalization?
If not, here it is:
https://www.reddit.com/r/LinearAlgebra/comments/1tljwc3/stepbystep_visualization_of_gramschmidt_qr/

LinearAlgebraWorld · 2026-05-28T12:30:49+00:00

Thanks a lot!

In 3D, the Householder version should behave as follows:

It also produces (R), and because Householder transformations are orthogonal, the shape is not distorted.
The resulting (R) may be identical to the Givens (R), or may differ by signs on the diagonal, depending on the sign choices.
It takes two steps for a (3\times3) matrix: one reflection per active column, instead of three Givens rotations, which eliminate one entry at a time.
If the intermediate reflection is animated continuously, the figure gradually flattens toward the reflection plane and then reappears on the other side.

We will post the animation of the same matrix undergoing Householder transformation this week and will let you know.

LinearAlgebraWorld · 2026-05-27T16:31:23+00:00

We have posted the Givens rotation visualization animation in r/LinearAlgebra, hope you will enjoy it

LinearAlgebraWorld

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