ABSTRACT
Ever since human thought evolved, understanding a physical phenomenon (as a system) and estimating the unknown components of the system has been an ongoing process. Reducing the study to a finite number of components (also called as parameters, which may be in thousands), establishing various linear relations between them and trying to find them is essential in almost all fields of Social, Financial, Physical, Biological, Mathematical Sciences and Engineering.
The concepts of a scalar, vector and a matrix help us in mathematically modelling the phenomena as a linear system consisting of the aforesaid entities. In order to deduce a meaningful conclusion from the linear system so obtained, the scalar, vector and a matrix must be understood and the operations that can be performed on them must be clearly known. With this intent this chapter concentrates on the basic structure and the related concepts.
As this is the first chapter of the book, the building blocks of all the topics covered in the book constitute this chapter. Section 1.2 begins with the definitions of a scalar, and a vector and proceeds with the study of their properties. These properties provide a base for the theory developed in later chapters. Matrices are introduced in terms of vectors in Section 1.3. This definition is useful as nearly in all applications of a matrix consisting of linearly independent vectors or ortho-normal vectors is utilized to transform a vector in a given situation. Various types of matrices which will be useful later are introduced in Section 1.4. Section 1.5 deals with elementary operations on matrices. These elementary operations are the only operations that can be done within a matrix and their importance is in developing algorithms for numerical techniques. Elementary matrices corresponding to each elementary operation are introduced in this section. Determinants, which attribute a numerical value to a matrix, are given along with their properties in Section 1.6. The contents of Section 1.7 consist of the inverse of a matrix together with its properties. Section 1.8 deals with the partitioning of a matrix, where in basic algebraic properties involving the partition are introduced. Section 1.9 introduces the concepts of pseudo inverse and modular inverse. Section 1.10 concludes the chapter.
