Optimal linear prognosis II. Geometry in space

Abstract

This article is the second part of the article "Optimal Linear Forecast". The results of this article continue the study of optimal linear projections in three-dimensional space from the point of view of comparing the lengths in two different metrics given by the original dot product and the second dot product of the first article. The introduction of a new metric leads to a situation where, from the point of view of linear dependence, it is necessary to change the unit of length depending on the direction of the vector of the observation plane. This change in length is impossible from the point of view of simple orthogonal transformations of the plane that transfer the sides of the rhombus into its diagonals, orthogonal in the case of both scalar products. From this point of view, it has been proven that the units of length measurement should be the same on the diagonals and sides of the rhombus, the sides of which coincide with the observation vectors in the symmetric case. In the symmetric case, the orthogonality of the diagonals of the rhombus leads to the same result, which, using the orthogonal transformation, must transform into the sides of this rhombus that are orthogonal in both metrics. The results related to the comparison of lengths on a plane lead to new evidence of the main provisions of the first article. These proofs use equalities similar to the Pythagorean theorem for the new metric. Since an arbitrary linear transformation on a plane coincides with the rotation of vectors through a certain angle, the results of the article require further study from the point of view of computational methods.