Truncated SVD

I'm a little confused about the dimensions of the matrices.
1) The Uk matrix has size DK, then the Uk^TU matrix must have size KD and after multiplying with the S matrix, the resulting matrix has size KN. But the Sk matrix has size D*N... And how was the last equality obtained - the matrix Uk replaced by the matrix U?
truncated SVD.JPG
2) Also, these 2 formulas seem contradictory to me. They are the same except for U and Uk.
3) Why multiplication by matrix U gives compression? Isn't Uk the compression matrix?
Thank you in advance!


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