This course is designed for students of mathematics department, and also for students who need more algebra, such as students in department of physics and department of computer science. It enables students to understand more fundamental contents of linear algebra, and to use algebraic methods to solve problems.After completing this course, students should be able to:1. Understand the definitions of vector spaces, linear maps and linear operators defined on various vector spaces.2. Understand the concepts of eigenvectors, eigenvalues of linear maps and invariant subspaces of vector spaces. 3. Have a deep understanding of properties of inner product spaces and their applications. 4. Understand the theory and applications of operators on inner product spaces, operators on complex vector spaces, and operators on real vector spaces.5. Understand the concepts of trace and determinant from the point of view of operators.