This textbook is designed to college sophomores and juniors the basics of linear algebra and the techniques of formal mathematics. There are no prerequisites other than ordinary algebra. The text has two goals: to teach the fundamental concepts and techniques of matrix algebra and abstract vector spaces, and to teach the techniques associated with understanding the definitions and theorems forming a coherent area of mathematics. There is an emphasis on worked examples and on proving theorems carefully. Contents: 1) Systems of Linear Equations. 2) Vectors. 3) Matrices. 4) VS Vector Spaces. 5) Determinants. 6) Linear Transformations. 7) Representations.