Possibly the Best Introductory Book on Linear Algebra

I recommend a globally popular undergraduate textbook, Linear Algebra and Its Applications. This book enjoys immense popularity abroad and is designated as a primary linear algebra textbook by many universities. It is also a required course book for various disciplines beyond pure mathematics, including but not limited to computer science, economics, and statistics. Some high school competition students also study this book in advance.

Linear algebra can be challenging; some students struggle to follow lectures (students abroad often complain about poor textbooks as well). This leads many to seek classic linear algebra textbooks online. The Chinese translation of this book is titled 《线性代数及其应用》, and it is also adopted by some universities in China.

The author, David C. Lay, holds a PhD in mathematics from the University of California and has many years of teaching experience in linear algebra (he began teaching at the University of Maryland in 1996). The first edition of this book was published in 1993. Many students find it easy to understand and read, appreciating how it intuitively introduces linear algebra concepts and expertly links abstract mathematical problems to practical applications. It is especially suitable for students outside pure mathematics. Of course, pure math majors tend to rate this book less favorably—different strokes for different folks.

Unlike many domestic textbooks, this book does not start with the “determinant” (some well-known Chinese textbooks begin with this topic). Instead, it first covers linear systems, then matrix algebra, and only later introduces “determinants” and “vector spaces.” In a previous response, I analyzed that mathematics textbooks must have a philosophical core. Studying math is not merely about scoring high marks. The “determinant,” as a technical tool, is not the starting point of linear algebra and does not need to appear at the beginning of the textbook. Of course, historical reasons contribute to the current practice—Soviet-style textbooks follow this approach, prioritizing exam readiness over genuine conceptual understanding, which seems to matter less.