Chapter 2

Matrices and Linear Transformations

Addition, Scalar Multiplication, and Multiplication of Matrices

Figure 2.1 An arbitrary m X n matrix A.

Figure 2.2 The elements of a square matrix A where the subscripts are equal, form the main diagonal.

Figure 2.3

Properties of Matrix Operations

Example 5

Figure 2.4

Figure 2.5

Example 6

Figure 2.6 Many networks are designed to accept signals at certain points and to deliver a modified version of the signals. Here, the usual arrangement is illustrated.

Example 6 (cont’d)

Figure 2.7 An example of a two-port.

Figure 2.8 Three two-ports with transmission matrices A, B, and C.

Exercise Set 2.2

Figure 2.9 Determine the transmission matrices of the two-ports.

Figure 2.10 A two-port that consists of three two-ports placed in series.

Matrix Transformations, Rotations, and Dilations

Figure 2.11

Figure 2.12

Figure 2.13 Rotation about the origin.

Figure 2.14 Matrix transformations.

Example 1

Figure 2.15

Figure 2.16

Figure 2.17

Theorem 2.9

Figure 2.18

Figure 2.19

Figure 2.20

Linear Transformations, Graphics, and Fractals

Figure 2.21 Here, the structure preserving ideas of a linear transformation are illustrated.

Figure 2.22

Figure 2.23

Example 5

Figure 2.24 Fractal pictures of nature.

A Communication Model and Group Relationships in Sociology

Figure 2.25

Figure 2.26

Exercise Set 2.9

Figure 2.27

Figure 2.28

Figure 2.29

Figure 2.30