Narrative

I was watching a Gilbert Strang lecture about the null space when I had an intuition.

Professor Strang just solved the following $3 \times 4$ matrix set equal to zero ($Ax = 0$) to find the null space. The matrix looked like this:

$\begin{bmatrix} 1 & 2 & 2 & 2 \\ 2 & 4 & 6 & 8 \\ 3 & 6 & 8 & 10 \end{bmatrix}$

and I noticed that after doing elimination, there were 2 pivot columns and 2 free columns. By setting the free columns to permutations of 1s and 0s, he found the two vectors that made up the null space:

c(-2, 1, 0, 0) + d(2, 0, -2, 1)

So I asked Gemini:

I feel like I’m on the edge of making a connection here… Seeing that made my mind jump all the way back to the first 3blue1brown linalg video where he paints the geometric picture of using 2 vectors to reach any point in $R^2$ […] This must relate back to the concept of rank somehow, does rank define the dimensionality of the null space?”

There were actually two great connections here, first the Rank-Nullity Theorem (The Fundamental Theorem of Linear Algebra). The math rule states: Total Columns = Rank + Nullity. Since your matrix had 4 columns, and a Rank of 2 (the independent info), the Nullity had to be 2 (the redundant info/free variables).

Second, my brain was reaching for orthogonality when it surfaced that initial 3blue1brown video from my memory. Stated a little better, I was wondering if all of the vectors that comprise the null-space are perpendicular. Which of course is not necessarily the case. I think the PCA stuff I was just working on had me focusing on that.

Anyway, the next morning Anthropic pushed out Opus 4.8. That fact coupled with a desire to centralize my notebooks in some modern solution on my server —instead of running them out of vscode on whatever machine I happened to be working on— resulted in me having Opus 4.8 spin up a nice marimo environment in a docker container on my app server.

Which brings me to

Moving on to predictions with the KNB dataset

I posted about the PCA work I did on the knoebels data already, that would be considered inference. Now, I wanted to see about creating a model with some predictive power for park busyness.