TITLE: Geometrical interpretations of SVD
QUESTION [9 upvotes]: I'm a bit confused by the various geometrical/visual interpretations of SVD or better I'm wondering how to reconcile them.

Transformations : As explained here, the 3 matrices produced by the SVD can be interpreted as rotation, scaling, rotation.
Projection on an axis (dimensionality reduction): As explained here, SVD enables to capture the most of the variance in the data by selecting the right orthogonal axes.

My (very naive) question is: are these 2 interpretations related visually or not at all, and if yes, how?  Or are they 2 unrelated applications of SVD?
As far as I understand, the transformations of the matrices (1) can be applied to anything/any object, so it's not about the data mentioned in (2) so I would say it's not related but I may be wrong (because in (2) we also talk about rotations...).
Please note that I'm looking for the intuition with as few math as possible (I don't know much linear algebra as you may have guessed).
Many thanks.

REPLY [2 votes]: Here's an interactive worksheet to help you understand (1)
https://www.geogebra.org/m/mrey8VJX