I am a little bit confused by a wording on the orthogonal projection given in my study guide for linear algebra (LSE).
It says "it is the projection onto S parallel to the orthongonal complement of S (for a particular subspace of S)."
I am confused by the world 'parallel'. If we take a simple case of a plane S and its orthogonal complement So (a line perpendicular to the plane), then any vector x can be described as a sum of two projections: x=x1+x2, where x1 - is the projection of vector x onto a plane S, and x2 - the projection of vector x onto the line So. The projection of the vector onto a plane S - x1 - is not going to be parallel to the line So - since it will belong to the plane S, it will be perpendicular to So.
So, looking back at the definition, what is 'parallel' in my example?
thanks a lot! (sorry I cannot figure out how to cut and paste math signs into the message text).
Choose . We have the decomposition:
and is the orthogonal projection of onto , and of course, is not parallel to .
Now choose for example , the projection onto of the point parallel to the orthogonal complement of is the intersection of (parallel affine variety to through ) with . In this case we obtain the point