Gauss-Jordan elimination

[davidaustinm]

I'm going to quibble about your presentation of the Gauss-Jordan elimination algorithm, in particular, Activity 1.2.14
https://library.tbil.org/2025/linear-algebra/LE2.html#LE2-4-14 and the video.

Your algorithm focuses on the pivot columns and produces 0's below and above the pivot position before moving on to the next column. The more traditional algorithm produces 0's below the pivot position only and then goes to the next column, which breaks the algorithm into two phases: forward substitution and back substitution. I see this as preferable for the following reasons:

1. It reduces flops and is how computers implement the algorithm. I realize that you're doing this "by hand" but a "flop" in this case represents student effort and the potential for mistakes (and this type of mental arithmetic already leads to lots of mistakes).
2. The upper triangular form obtained from forward substitution with less effort reveals a great deal about the matrix, namely its rank, a basis for the column space, the determinant, and more. We often don't care about the full RREF.
3. Your implementation will not produce an LU factorization whereas LU is a natural byproduct of the traditional algorithm.
4. Pedagogically, this also allows for better scaffolding and motivation. Clearly, a diagonal system is the easiest. Starting with an upper triangular system, we can work right to left to get a diagonal system so it turns out that's pretty easy too. So how can we get a general system into an upper triangular form? Implement the forward substitution pass.

#teamforwardsubstitution

[StevenClontz]

Hmm, neat, I'd be in support of such a refactor for 2026 edition. Maybe I'll try to get this in before I teach LE2 later this week (or early next week). What say you @jkostiuk ?

[jkostiuk]

i'm a fan, particularly if we then seek opportunities to revisit the LU decomposition---even if this is mostly relegated to the mathematical writing explorations. i think it'll improve the lesson and then give us some additional content that is then pretty accessible.

[StevenClontz]

Alright, I'll work on a PR today

[siwelwerd]

I kind of like it how it is, for various reasons, but I am not teaching this anytime soon so would be amenable to folks trying it out and basing a decision on your reports of student feedback rather than my speculation.

[jkostiuk]

i'm also curious on what your students think if you test(ed) it out, steven.

An alternative way to incorporate some of dave's feedback would be to do a math writing exploration about forward/backward substitution and linking up with LU-decompositions on the MX module.

[StevenClontz]

Still working on #856 and will use it this week, likely tomorrow.