14 - Analytic Geometry: Part 4 — The Geometry of Vectors and Matrices

Abstract

This chapter illustrates the geometry of vectors and matrices. It discusses a row matrix M in column space. The row vector M projects on the plane defined by the pre-defined columns as a point or a vector (straight line) with a direction angle. A matrix consisting of more than one row can be represented by 2-D row space. Each column in the matrix can be represented by a column vector. The principal components for regression vectors are demonstrated from the projection of two column vectors—C1 and C2—on their vector sum or principal component (PC1). The vector sum of the two column vectors passes through a definite point, but the projection of each column on PC1 gives a vector with a length equal to the sum of the line segments. By representing a row vector in column space or a column vector in row space, the geometry of regression can be effectively illustrated. These concepts combined with matrix algebra are useful for the further discussions of regression.