This book contains a brief course in elementary linear algebra with an emphasis on solving systems of linear algebraic and ordinary differential equations. It is written for undergraduate students in applied mathematics, science and engineering. Basic knowledge of the arithmetic of complex numbers and exposure to elementary functions and calculus are assumed.

Chapter 1 covers the basics of matrices and vectors, giving definitions and concepts needed in linear algebra studies in later chapters.

Chapter 2 is concerned with solving systems of linear algebraic equations. It shows how elementary row operations on an array of numbers can be used to reduce a given system of linear algebraic equations to a simpler but equivalent system that can be easily solved. The chapter also introduces the concept of linearly independent vectors and explains how the task of determining whether a given set of vectors is linearly independent or not can be formulated in terms of a homogeneous system of linear algebraic equations.

Chapter 3 examines elementary and invertible matrices. It shows how elementary row operations can be performed on an invertible square matrix to find its inverse matrix and explains how matrix invertibility is related to solving a system of linear algebraic equations. Formulae for some properties involving inverses of matrices are given in the chapter.

Chapter 4 begins with a formula defining the determinant of a square matrix, shows how elementary row operations can be performed on a matrix to calculate its determinant and derives alternative formulae for the determinant. The relation between matrix determinant, matrix inverse and solutions of systems of linear algebraic equations is explained.

Chapter 5 deals with the matrix eigenproblem and the matrix diagonalization problem. The two related problems are of fundamental importance in linear algebra. The chapter explains how they can be applied to solve homogeneous systems of first order linear ordinary differential equations.

Chapter 6 gives a summary of the definition of terms and the main results in the earlier chapters.