Professors Wanmo Kang and Kyunghyun Cho have reimagined how linear algebra is taught to students in the era of data science and artificial intelligence. They realized that key concepts like projection, singular value decomposition (SVD), and positive definiteness are frequently used in practice but often overlooked in traditional linear algebra courses. These courses focus on invertibility and mathematical derivations rather than practicality and usefulness.
To address this issue, Kang and Cho have written a new textbook on linear algebra that introduces concepts in a radically different order. The book covers matrices, vector spaces, orthogonality, and SVD early on, without compromising mathematical rigor. After reading just the first few chapters, this approach allows students to become proficient in useful results and algorithms from linear algebra.
The textbook has already been used at the Korea Advanced Institute of Science and Technology (KAIST) and has received feedback that has impacted its organization. The authors believe that their problem-driven approach will prepare students to tackle real-world questions in data science and artificial intelligence.
Rethinking Linear Algebra Education for Data Science
The advent of data science and artificial intelligence has prompted a reevaluation of how linear algebra is taught to students. Professors Wanmo Kang and Kyunghyun Cho have been discussing this topic for years, leading to research collaborations and the realization that certain concepts in linear algebra are frequently invoked in practice, but often overlooked or underemphasized in traditional courses.
The Importance of Practicality
The professors noticed that existing linear algebra courses tend to focus on mathematical derivations and introductions rather than practical applications. This approach can lead to students being introduced to concepts out of order, with more emphasis placed on invertibility and less on practically useful topics like projection, singular value decomposition (SVD), and positive definiteness. The authors argue that a radically different approach is needed, one that prioritizes practicality and usefulness.
A New Textbook for Linear Algebra
To address this issue, Professors Kang and Cho have created a new textbook on linear algebra, which reimagines the way concepts are introduced and taught. The book ensures that useful and practical materials are not relegated to the end of the book but rather treated early on, without compromising mathematical rigor. This approach allows students to become proficient in various results and algorithms from linear algebra even after reading just a few chapters.
A High-Level Table of Contents
The textbook’s table of contents provides a sense of what this new approach entails:
- Introduction
- Matrices and Gaussian Elimination
- Vector Spaces
- Orthogonality and Projections
- Singular Value Decomposition
- SVD in Practice
- Positive Definite Matrices
- Determinants
- Further Results on Eigenvalues and Eigenvectors
- Advanced Results in Linear Algebra
- Big Theorems in Linear Algebra
Notably, the singular value decomposition (Chapter 5) is placed earlier than the determinant (Chapter 8) and the eigendecomposition (Chapter 9), reversing the typical order of topics in linear algebra books.
An Optimization/Variational Approach
The authors adopt an optimization/variational approach to solve problems, formulating projection and singular value decomposition. This approach is particularly well-suited for data science and artificial intelligence applications, where many questions involve natural optimization. By adopting this problem-driven setting, the authors believe that linear algebraic tools obtained through this approach will be better equipped to tackle real-world questions.
Additional Chapters in Appendix
The textbook includes several chapters in an appendix, covering concepts and results that are not directly about linear algebra but are necessary for deriving various results in linear algebra. These topics include:
- Convexity
- Permutation and its Matrix Representation
- The Existence of Optimizers
- Covariance Matrices
- Complex Numbers and Matrices
- An Alternative Proof of the Spectral Decomposition Theorem
Real-World Applications and Feedback
The textbook has already been used in a course at the Korea Advanced Institute of Science and Technology (KAIST)’s Department of Mathematical Sciences, with feedback from students greatly impacting the textbook’s organization.
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