Pages: 17

This chapter discusses the classification, properties of matrices, and matrix operations needed to study the solution of linear equations and elementary matrices. It introduces the operations of addition and multiplications for matrices and defines the algebraic properties of these operations. The powers of matrices and inverses of matrices are also described. These tools lead to different methods for solving linear systems and insights into their behavior. It lays the foundation for using matrices to define functions called linear transformations or vector spaces. The reader will identify how matrices are used in various applications. They are applied in archaeology to determine the chronological order of artifacts, cryptography to ensure security, and demography to predict population movement. The inverse of a matrix is used in a model for analyzing the interdependence of economics. Wassily Leontief received a Nobel Prize for his work in this field. This model is now a standard tool for investigating economic structures ranging from cities and corporations to states and countries. Throughout these discussions, we shall be conscious of numerical implications in finding the need for efficiency and accuracy in implementing matrix models.

Pages: 40

This chapter discusses the system of linear equations and its solution using the Gaussian elimination method, Gauss-Jordan elimination method and LU decomposition methods, the definitions of determinants, their properties, and Crammer’s rule. Linear algebra is a branch of mathematics that plays a central role in modern mathematics and is essential to engineers and physical, social, and behavioral scientists. When mathematics is used to solve a problem, it often becomes necessary to find a solution to a so-called system of linear equations to study methods for solving such equations. This chapter introduces methods for solving linear equations and looks at some of the solutions’ properties. It is essential to know the solutions to a given system of equations and why they are the solutions. If the system describes some real-life situation, then understanding the behavior of the solutions can lead to a better understanding of the circumstances. Associated with every square matrix is a number called its determinant. The determinant of a matrix is a tool used in many branches of mathematics, science, and engineering. At the same time, the method of Gauss-Jordan elimination enables us to find the inverse of a specific matrix. For example, it does not give us an algebraic formula for the inverse of an arbitrary matrix, a formula that can be used in theoretical work. Determinants provide us with such procedures. Furthermore, criteria for when specific systems of linear equations have unique, none, or many solutions can be stated in terms of determinants. In this chapter, the determinant is defined, and its properties are developed.

Pages: 36

This chapter introduces the notion of an abstract vector space with its typical features. Vector spaces occur in various branches of mathematics and have many applications in science and engineering. In this chapter, we generalize the concept of vector space Rn and examine its underlying algebraic structure. Any set with this structure has the same mathematical properties as Rn and will be called a vector space. The results that were developed for the vector space Rn will also apply to such spaces. We shall, for example, find that specific spaces of matrices and functions have the same mathematical properties as the vector space Rn. The concepts of linear independence, spanning set, basis, linear transformations, and dimension will be extended to these spaces.

Pages: 20

This chapter discusses the properties of eigenvalues and eigenvectors along with some properties of inner product spaces. It also includes the Gram−Schmidt orthogonalization process and QR-factorization. Eigenvalues and eigenvectors are particular scalars and vectors associated with matrices. They can be expressed in terms of determinants. Many numerical methods make use of eigenvalues and eigenvectors. We shall use eigenvalues and eigenvectors to make long-term predictions of populations in analyzing oscillating systems and in studying the connectives of road networks.

Pages: 39

This chapter discusses an integral part of this subject, i.e., linear trans-formations, including the range and Kernel of transformation, one-to-one and invertible transformation, coordinate representation, change of basis, isomorphism, matrix of linear transformations, visualization of the matrix representation, and relation between matrix representation. The significance of vector spaces arises from the fact that we can pass from one vector space to another by means of functions that possess a particular property called linearity. These functions are called linear transformations. This chapter aims to familiarize the students with the fundamental properties of linear transformations. A topic such as a Kernel, range, and rank nullity theorem are presented here. Linear transformations, range and Kernel, are used to give the reader a geometrical picture of the sets of solutions to systems of linear operations, both homogenous and non-homogenous. The rank nullity theorem and its consequences are presented in detail. The theory developed so far is applied to operator equations and, in particular, to differential equations. This application discloses that the solution space of the nth-order homogenous linear ordinary differential equations is an n-dimensional subspace of n-times continuously differentiable functions.

Pages: 31

This chapter discusses the generalization process of the algebraic properties to Rn, i.e., extending the concepts of dot products of two vectors to the norm, the angle between vectors, and the distance between points in general vector spaces. This study will enable us to discuss the magnitude of orthogonal functions and approximate functions by polynomials, a technique used to implement functions on calculations and computers. It will no longer be restricted to Euclidean geometry and create geometries on Rn, such as introducing the geometry of special relativity and discussing the implication of this theory for space travel by looking at a voyage to the star Alpha Centauri.

Pages: 16

Matrices form a vital tool in the study of finite-dimensional vector spaces. Hereafter though, for convenience, we deal with real vectors and the real vector space Vn, all our definitions, unless otherwise restricted, will also apply to complex vectors and the complex vector space VnC. We have developed much mathematics around the concept of a basis. In this chapter, we discuss the relationship between coordinate systems and bases. We have found that a linear transformation can be represented by a matrix relative to a standard basis. We shall find a matrix representation relative to every basis. If possible, it will be of interest to find diagonal representations and determine the basis (or coordinate systems) to which this applies. Eigenvalues and eigenvectors play an essential role in these discussions. This technique will enable us to find the most suitable coordinate systems for discussing physical situations such as vibrating strings. We shall define remarkable structure-preserving transformations between various types of vector spaces. We shall find that even though the elements of specific vector spaces such as Rn and Pn differ in appearance, their mathematical properties have much in common. It means that any results we develop for Rn can be applied to all such vector spaces.

Pages: 32

This chapter discusses the diagonalization process that includes minimal polynomials, Cayley−Hamilton theorem, and diagonal matrix representation of a linear operator. It also consists of the diagonalization of matrices, diagonalization of symmetric matrices, and orthogonal diagonalization. A self-contained illustration of the role of linear transformation in computer graphics is presented. The bases of the space define these isomorphisms. Different bases also lead to other matrix representations of linear transformation. The vital aspect of eigenvalues and eigenvectors and minimal polynomials in finding diagonal representations is discussed. These techniques are used to arrive at the normal modes of oscillating systems.

Pages: 25

This chapter discusses the applications of conics and quadrics and the canonical form of matrices that include some vital discussion such as bilinear forms, Sylvester’s law of inertia. In this chapter, we shall use the concept of an orthogonal matrix to reduce a real symmetric matrix to the diagonal form by a similarity transformation. This chapter starts with the discussion of quadratic forms, conics, and quadrics. It includes some vital discussion such as bilinear forms, eigenvalues of congruent matrices, Sylvester’s law of inertia, and skew-symmetric bilinear form.

Pages: 25

This chapter discusses the applications of special matrices such as block diagonal and triangular matrices, symmetric and Hermitian matrices, Schur’s theorem, spectral theorem, Jordan and rational canonical form of special matrices. This chapter starts with some special types of matrices, including some essential theorems such as Schur’s theorem, spectral theorem, and some crucial canonical and normal forms such as Jordan canonical form rational canonical form and minimum polynomial of Jordan normal form.

Pages: 25

This chapter discusses the approximation of functions, least square problems having their vital application in numerical analysis. The chapter closes with a discussion of the approximation of functions and the use of pseudo inverse to determine least square curves for given data.