MAO1103: Linear Algebra

Introductory
  1. Introduction to linear transformations
  2. Linear sums and independence, span, and linear bases
  3. Why matrices?
  4. Composition of linear transformations: matrix multiplication
  5. Inverses, determinants, column spaces, non-square matrices
  6. Null and row spaces, transpose and the dot product
  7. Basis changes
  8. What does it mean for two matrices to commute?
  9. Eigen-everything, diagonalisation, repeated eigenvalues
  10. Generalised eigenvectors, Jordan normal form
  11. An introduction to forms
More
  1. Trace and its links to determinant and eigenvalues
  2. Some interesting theorems in linear algebra
  3. Complex linear algebra
  4. The Hilbert space
  5. Some matrix theory
  6. Affine transformations
Insight into cross products series
  1. Quaternions introduction: Part I
  2. Quarternions introduction: Part II
  3. Octonions and beyond
  4. Exterior products (aka: a primer on tensors)
  5. That silly formal determinant
Reference
  • Computational techniques (proofs regarding direct formulae for determinants, inverses)
  • The geometry of $\mathbb{R}^n$ -- cross product, planes, projections, 3D rotations, etc.
Resources
  • The Essence of Linear Algebra youtube video series (3Blue1Brown) [insight]
  • Khan Academy Linear Algebra [companion]
  • Schaum's outlines: linear algebra [companion]
  • Calculus II (Apostol) -- chapters 1-5 [companion]
  • Linear Algebra done right (Axler) [textbook]
  • Linear Algebra done wrong (Treil) [textbook]
  • Introduction to linear algebra (Lang) [textbook]
  • A primer on linear algebra (Herstein) [textbook]
  • Introduction to Linear Algebra (Strang) [textbook]
  • Linear Algebra (Lang) [textbook2]
  • Matrix theory and linear algebra (Herstein) [textbook2]
  • Linear Algebra and its applications (Lax) [textbook2]
  • Linear Algebra and its applications (Strang) [textbook2]

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