Textbook in PDF format

This textbook is intended as an aid for students who are attending a course on Linear Algebra in universities and colleges. It is suitable for students who are tackling Linear Algebra for the first time and even those who have little prior knowledge in mathematics. This is a self-contained textbook that contains full solutions to the questions in the exercises at the end of the book. The aim is to present the core of Linear Algebra in a clear way and provide many new and simple proofs for some fundamental results. For example, we give a very simple proof for the well-known CauchySchwarz inequality. The goal is to enhance students’ abilities of analyzing, solving, and generalizing problems. The material covered in this textbook is well organized and can be easily understood by students. The main topics include vectors, matrices, determinants, linear systems, linear transformations in the Euclidean spaces, bases and dimensions of spanning spaces in the Euclidean spaces, eigenvalues and diagonalizability, subspaces of the Euclidean spaces, and general vector spaces. When we discuss linear transformations, bases, and dimensions, we restrict our attention to the Euclidean spaces, but the ideas and methods involved can be extrapolated to general vector spaces. This textbook is based on the third edition of Linear Algebra. Some material has been rewritten for greater clarity and some sections and chapters have been combined for better understanding. New examples and exercises have been added. One new chapter introducing vector spaces has also been added. In particular, subspaces of the Euclidean spaces are discussed in more detail