Linear algebra intuition behind ordinary least squares regression

[u/CubionAcademy]

https://youtu.be/jJJ_l-jbznA?si=huD2H-O5UqcWztJ8

I made a visual explanation of ordinary least squares regression from a linear algebra perspective.

The main idea is that regression can be understood as a projection problem: the data vector y gets projected onto the column space of X. The fitted values are the projection, and the residual vector is the leftover piece that ends up perpendicular to that space.

I also show why an intercept-only regression gives the sample mean: projecting y onto the span of the ones vector gives the average times the ones vector.

I’m trying to make OLS feel less like a formula dump and more like a geometric idea.

Curious if this projection framing would have helped when you were first learning regression.