What alien language did he just say?

indeed, this man is single-handedly responsible for me understanding this and everyone else explains it like its defense against the dark arts

It took me some time to figure it out, but the idea is that we simply take a projection of X_p+1 onto X_p, so X_p transposed is just a part of the projection formula. The idea of the projection is such that you find some coefficient, which when multiplied by X_p, would produce such a scaled version of X_p that is the closest to X_p+1. We know that when p is sufficiently large, both X_p+1 and X_p are simply some multiples of e1, so they are collinear, which means X_p+1 can be expressed as X_p multiplied by some coefficient and that is the coefficient that projection gives us. Now if we look at the formulas of X_p+1 and X_p, the difference between them is just an extra multiple of lambda, so lambda = projection coefficient of X_p+1 onto X_p.

As far as i understand, he says that the col vectors can be written as lin combin of evectors, as it spans Rn.

pretty sure he’s writing normally then when he edits the video he mirrors it so the writing is facing us

Haha!

I think it's more like he has the glass pane between himself and the camera and the camera simply reflects everything he writes.