Introduction:
The linear algebra section of the UPSC Mathematics Optional subject exam covers a wide range of topics, including vector spaces, linear transformations, eigenvalues and eigenvectors, inner product spaces, and matrix theory. To excel in this section, it is crucial to have access to the best study materials that provide comprehensive coverage of the topics and offer practice questions and solutions. Here are some recommended study materials for the linear algebra section of the UPSC Mathematics Optional subject exam:

1. Linear Algebra by Hoffman and Kunze:
- This is a widely acclaimed textbook that covers all the essential topics of linear algebra in a rigorous and comprehensive manner.
- It provides a detailed explanation of concepts, proofs, and examples to help students understand the subject deeply.
- The book also offers a wide range of exercises with solutions to practice and reinforce the learned concepts.

2. Linear Algebra by Schaum's Outline Series:
- This study guide is known for its concise and clear presentation of the subject matter.
- It provides an overview of the key concepts, definitions, and theorems of linear algebra, making it suitable for quick revision.
- The book also includes numerous solved problems and practice exercises with step-by-step solutions to enhance problem-solving skills.

3. Linear Algebra Done Right by Sheldon Axler:
- This book focuses on the fundamental concepts of linear algebra and emphasizes the importance of understanding the underlying theory.
- It provides a fresh and intuitive approach to the subject, making it accessible to students with different mathematical backgrounds.
- The book includes a variety of exercises at varying levels of difficulty, enabling students to practice and master the concepts.

4. Linear Algebra and Its Applications by David C. Lay:
- This textbook offers a balanced approach to the subject, combining theoretical concepts with practical applications.
- It covers a wide range of topics, including vector spaces, linear transformations, matrix algebra, and eigenvalues and eigenvectors.
- The book includes numerous real-world examples and applications, helping students relate the concepts to their practical significance.

5. Previous Years' Question Papers:
- Solving previous years' question papers is an essential part of exam preparation.
- It helps in understanding the exam pattern, time management, and identifying the important topics.
- UPSC Mathematics Optional subject exam question papers from previous years can be found on the official UPSC website or other reliable sources.

Conclusion:
To excel in the linear algebra section of the UPSC Mathematics Optional subject exam, it is essential to have access to comprehensive study materials that cover all the topics and provide ample practice opportunities. Textbooks like "Linear Algebra" by Hoffman and Kunze, "Linear Algebra" by Schaum's Outline Series, and "Linear Algebra Done Right" by Sheldon Axler are highly recommended for their detailed explanations, examples, and practice exercises. Additionally, solving previous years' question papers helps in familiarizing oneself with the exam pattern and identifying important topics.