Introduction

Linear algebra is a fundamental subject in mathematics with applications in various fields. Among the many texts available, Sheldon Axler's Liner Algebra Done Right has gained significant popularity among both educators and learners. This article explores mathematicians' views on Axler's approach, highlighting both the praise and criticism from the perspective of conceptual clarity, pedagogical innovation, and the focus on proofs. Additionally, we examine the arguments against his stance on determinants, drawing from the experiences and opinions of a mathematics student who learned from the book.

Praise for Axler's Approach

Conceptual Clarity

One of the most significant benefits of Axler's book is its emphasis on conceptual clarity. Unlike many traditional texts that rely heavily on computational techniques and determinants, Axler focuses on vector spaces and linear transformations. This approach allows readers to develop a deeper understanding of the underlying concepts, making the learning process more intuitive and effective.

Pedagogical Innovation

Axler's organization of the material is often seen as more innovative than traditional texts. By postponing the introduction of determinants, he enables students to first build a solid foundation in vector spaces, linear transformations, and eigenvalues. This pedagogical strategy is widely regarded as more fundamental to the subject, leading to a more robust and comprehensive understanding of linear algebra.

Focus on Proofs

Axler's text encourages students to engage with proofs and theoretical aspects of linear algebra, which is essential for developing mathematical maturity. By emphasizing the importance of rigorous proofs, the book prepares students for more advanced mathematics and research.

Criticism of Axler's Stance on Determinants

Determinants as a Tool

Some critics argue that while determinants are sometimes overemphasized in traditional courses, they remain an important concept in linear algebra. Determinants provide valuable insights, particularly in understanding properties of linear transformations such as invertibility and volume scaling. Removing determinants entirely may lead to a lack of exposure to these crucial computational techniques, which are essential in applications of linear algebra, especially in fields like engineering and physics.

Pedagogical Balance

Critics also suggest that completely sidelining determinants can lead to a lack of exposure to important computational techniques. This may result in a pedagogical imbalance, where theoretical concepts are prioritized at the expense of practical skills. Some educators and students may prefer the traditional approach, which integrates determinants throughout the course. They argue that this can help students learn how to compute and apply determinants in various contexts.

Student Perspective: The Role of Determinants

From a student's viewpoint, particularly someone who learned linear algebra based on Axler's text, there is significant value in delayed introductions. A Harvard Math 55A class, which I took and was based on Axler's book, provided an excellent introduction to linear algebra. I completely agree with his approach, especially in understanding the role of determinants and the importance of rigorous definitions.

Defining determinants at the very end, as Axler does, aligns with the book's overall focus on conceptual clarity. However, the definition of a determinant can be quite complex, requiring several important constructions. Even if rigour is not always considered a sufficient reason to justify its delayed introduction, it is essential to explain why the determinant is independent of a choice of basis. This explanation helps solidify the understanding of why the determinant is a fundamental concept in linear algebra.

Conclusion

The debate about Axler's approach to linear algebra is ongoing. While his focus on vector spaces and linear transformations is commendable, there are valid reasons for considering the role of determinants. The choice between these approaches ultimately depends on the specific goals and audience of the course. Regardless of one's stance, Axler's contributions have significantly impacted the way linear algebra is taught and learned.