Why dont we perform step 5. in the assignment?

Course Q&A Math for Machine Learning Linear Algebra for Machine Learning and Data Sc...
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There are two ways you can do it:

  1. Create the projection matrix first with the reduced number of columns (step 5 above) and then dot the centered data with that.
  2. Do the dot product with the complete eigenvector matrix and then “slice” the columns of the resultant matrix to create the reduced output.

The result is the same either way. You’re right that it would be more efficient to do it in the order described in 1).

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Or if your question is why we don’t divide the columns of V by their norms, it turns out that is not necessary when we write the code using scipy.sparse.linalg.eigsh. You can check and confirm that it returns the eigenvectors as unit vectors (vectors of length 1).