The first course in linear algebra is written by two experts from Algebra who have more than 20 years of experience in Algebra, linear algebra and number theory. It prepares students with no background in linear algebra. After mastering the materials in this textbook, students can already understand any linear algebra used in more advanced books and research papers in Mathematics or other scientific disciplines. This book provides a solid foundation for the finite-dimensional vector spaces theory. It explains in detail the relation between linear transformations and matrices. One may thus use different viewpoints to manipulate a matrix instead of a one-sided approach. Although most examples are for real and complex matrices, a vector space over a general field is briefly discussed. Several optional sections are devoted to applications to demonstrate the power of linear algebra.