Regression is Projection

Great Video. This actually helped a lot with my intuition in linear algebra 3.

Great video man

Great work. Please do more video with black background it’s better to watch at night

hello, economist here, incredible, it solves a lot of questions I had but was too lazy to look up. I really should get back into maths.

Absolutely loving the channel! Please keep it up. Can't wait to see the next ones

Excellent video!!!

Make more videos!

excellent. the last part was really impressive.

Could you maybe go through the logic of AA^Tv (or the full A(A^TA)^-1A^Tv 👀 from the perspective of your transpose video? So far I have that A is a linear map from the component-space of your sub space into real space (where x,y,etc., v live), and that A^T maps v into that component space in a dimension-removing manner (since rank(A) is the same as rank(A^T)), and thus it is that first multiplication by A^T that really does the dimension collapse of the projection. What I’m struggling with is why the multiplication by A gives us the result we want - it feels like there’s some fundamental dot-product relationship between x, y, v, maybe v-p, and p that should dictate why the dot-product-preserving nature of A vs A^T should matter but I don’t think I fully internalized the first projection matrix video enough to understand what it is.

OH and I guess that means (A^TA)^-1 is doing some kind of operation in that component space, I guess one that cancels out the non-orthonormality of A? But without understanding why the components make sense in the first place I don’t know if I can think it through. Might be the subject of a cool ellipse animation though like you did with the SVDs of A / A^-1^T in the transpose video

tysm, seeing the video makes the idea so clear

When the next video will be out, can't wait although I don't need it for my course I want to learn.

Your videos are incredibly good. A binged your channel. I wish you made more videos! May I suggest explaining a principle component analysis to further expand on projection and least squares solutions?

What you said about dot products not changing after projection, that was such an eye opener. I really hope you can do principle component analysis

Excellent explanation, especially the least squares part!

Great video. Very insightful.