This book evolved from our lectures in the advanced algebra courses at Nanjing University. It is intended for use by instructors and undergraduate students in a one-year advanced algebra course.The topics covered in this book consist of integers and polynomials, determinants and matrices, linear systems, linear spaces, linear maps, λ-matrices, quadratic forms, inner product spaces, and bilinear forms.There are sufficient well-selected exercises of a wide range to provide ample practice, expand coverage of topics treated in the text, and challenge the strongest students.Our objective is to prepare students with a solid foundation for pursuing advanced mathematics. We hope that the material presented here will interest the students and help them ask their own questions, look for their own examples, and discover their own proofs.