Several books on linear algebra have been written however, the present rigorous and transparent introductory text can be used directly in class for students of applied sciences. In an effort to bring the subject to a wider audience, this book provides a compact but thorough introduction to the topic. This book is intended for a senior undergraduate and for a beginning graduate one-semester course. The subject matter has been organized in the form of theorems and their proofs, and the presentation is rather unconventional. It comprises 25 class tested lectures that the first author gave to math majors and engineering students at various institutions over a period of almost 40 years.