After an explanation of the notation, we find,
We could then construct for each score. If we squared each and summed over all the scores, we would obtain
Obtaining this second equation from the first seems remarkable, but the textbook offered no insights on how one could see this.
Here's one possibility.
If we think of vectors , , and , then the second equation above is the assertion that
This equation is true whenever the vectors and are orthogonal.
But is precisely the orthogonal projection of onto the space spanned by and (see my earlier blog), so is in that space and is in the orthogonal complement.