Linear Algebra Done Wrong

 Sergei Treil

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Brown University has two introductory linear algebra courses. This text is used in the honors course that emphasizes proofs. The book’s title suggests that it is not the typical approach to linear algebra even among those books that are more theoretical.

For example, the concept of a basis is treated as more fundamental than the concept of linear independence, and linear transformations are introduced before solving systems of linear equations. Especially noteworthy is the motivation and development of determinants. As the author states in the preface:

I spent a lot of time presenting a motivation for the determinant, and only much later give formal definitions. Determinants are introduced as a way to compute volumes. It is shown that if we allow signed volumes, make the determinant linear in each column,… and assume some very natural properties, then we do not have any choice and arrive at the classical definition of the determinant.

Table of Contents

  1. Basic notions
  2. Systems of linear equations
  3. Determinants
  4. Introduction to spectral theory (eigenvalues and eigenvectors)
  5. Inner product spaces
  6. Structure of operators in inner product spaces
  7. Bilinear and quadratic forms
  8. Dual spaces and tensors
  9. Advanced spectral theory