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Figure 3.3 A point observed in two different reference frames.

      As seen in Figure 3.3, the position vectors of P are related to each other as follows:

(3.165)

Equation (3.165), which is a vector equation, can be written as the following matrix equation in one of the involved reference frames, say .

(3.166)

However, it is more convenient to express in rather than in . By doing so, Eq. (3.166) can be written again as follows:

(3.167)

      3.9.2 Homogeneous, Nonhomogeneous, Linear, Nonlinear, and Affine Relationships

      Consider two column matrices and . Suppose they are related to each other by means of a function so that

(3.168)

Depending on the mathematical features of the function , the relationship described by Eq. (3.168) is characterized by various designations, which are explained below.

      The relationship set up by is called homogeneous if

      (3.169)

      It is called nonhomogeneous if

      (3.170)

      The relationship set up by is called linear if, for a scalar k and for all Скачать книгу