Linear Algebra

Lecture 1. Matrices and Linear Systems. Algebra of Matrices

Lecture 2. Linear Systems

Lecture 3. The Inverse Matrix

Lecture 4. Coordinate Space ℝⁿ. Linear Combinations and Bases

Lecture 5. Rank of a Matrix

Lecture 6. Properties of the Rank. Solution Set of a Linear System

Lecture 7. Interlude: Permutations

Lecture 8. Determinant of a Square Matrix

Lecture 9. Properties and Applications of Determinants

Lecture 10. Cramer's Rule and Vandermonde Determinant. Interlude 2: Algebraic Structures

Lecture 11. Interlude 3: Complex Numbers

Lecture 12. Vector Spaces and Subspaces

Lecture 13. Sum and Intersection of Subspaces

Lecture 14. Quotient Space. Coordinates of a Vector

Lecture 15. Linear Mappings

Lecture 16. Space of Linear Mappings. Dual Space

Lecture 17. Affine Spaces

Lecture 18. Lines and Planes in Affine Space. Algebras, Lie Algebras, and Modules

Lecture 19. Modules over Rings. Linear Operators

Lecture 20. Projection and Reflection Operators. Polynomials of Linear Operators

Lecture 21. Realisation and Complexification. Invariant Subspaces of Operators

Lecture 22. Eigenvalues and Eigenvectors

Lecture 23. Diagonalizable and Nilpotent Operators. Spectrum of an Operator

Lecture 24. Normal Form of a Nilpotent Operator

Lecture 25. Jordan Normal Form

Lecture 26. Applications of Jordan Normal Forms. Normal Forms for Operators in Real Vector Spaces

Lecture 27. Matrix Functions