Geometric Interpretation of Ordinary Least Squares: An Introduction 

2:59 The space described in the video is 4D, albeit drawn on a 2D graphic tablet. So 3 independent vectors in 4 dimensional space do span a "plane".

@@kottelkannim4919 yeah! Thats a hyperplane (sounds much fancier than it is)

@@kottelkannim4919 n

no, u hat is the distance between the orthogonal projection of y over the space (in this example represented as mu hat on a plane) spanned (obtained through linear combinations) by the columns or vectors (column space) and the actual y. there is a unique linear combination obtained through the Ols method that minimizes the value of u hat. the parameters of such "optimal" combination are the betas of the regression. u hat can be also interpreted as the portion of y that cannot be expressed through a linear combination of the columns (i.e. the part of y that does not lie on the column space). why is u hat perpendicular to the row space? due to the fact that u hat lies in the so called "left null space", which as stated in the fundamental theorem of linear algebra is orthogonal to the column space.

Agreed; there's also another error: the vector in the column space should be XB (in the graph now it's u) and the difference vector should be y-XB = u (instead of y-u as in the video 4:54)