Program of the course on Linear Algebra

Abdus Salam School of Mathematical Sciences at the CG University, Lahore, Pakistan.
First semester. Professor Alexei Stepanov

1. Linear space: definition and examples.
2. The Nim game.
3. Subspace. Span. Linear dependence. Generating sets. Basis. Zorn Lemma.
4. Making a basis from a linearly independent or generating set. Basis-change matrix. Dimension. Isomorphism theorem. HA1.
5. Examples of basis choice. Finalization of Nim game solution.
6. Dimension of sum and intersection of subspaces. Direct sum (inner and outer).
7. Exercises
8. Linear maps. Matrix of a linear map. Exercises.
9. CT1. Transformation of a matrix of a linear map under a basis change. Image, kernel, invariant subspaces. Rank of a linear map.
10. Dimension of the image and the kernel. Invariants of linear operators: rank, determinant, and trace.
11. Eigenvalues and eigenvectors. Characteristic polynomial. Diagonal form. Algebraic and geometric multiplicities. HA2
12. Exercises. CT2.
13. Definition of the Jordan canonical form. Application: matrix equations. HA3.
14. Algebra of operators. Kernel of a polynomial at an operator.
15. Quotient space. Triangulation of an operator. Cayley–Hamilton theorem.
16. Nilpotent operators, Jordan chains. Proof of the Jordan theorem.
17. Exercises. HA4.
18. Bilinear and quadratic forms over an arbitrary field. Matrices of a bilinear form in differnt bases. Quadratic spaces. Diagonalization of a quadratic form: Lagrange method.
19. Euclidian and Hermitian spaces. Cauchy–Schwarz inequality, triangle inequality. Gram–Schmidt orthogonalization. Linear independence of an orthogonal set.
20. Exercises.
21. Orthonormal basis. Coordinates in an orthonormal basis. Isomorphism theorem for Eucledian and Hermitian spaces. Orthogonal decomposition. Parseval's identity and Bessel's inequality.
22. Overdefined linear systems: least squares approximation. HA5.
23. Dual space. Hom(V,V^*) and the second differential. Dual operator. Identification of the space with its dual via an inner product. Self dual operators.
    {Symmetric bilinear forms}={Quadratic forms}={Self-dual operators}={Symmetric matrices}.
24. Eigenvalues and eigenvectors of a self-dual operator. Orhtogonal (Hermitian) operators, isometry. Orthogonal (Hermitian) matrices.
25. CT3. Diagonalization of a quadratic form by an orhtogonal (unitary) transformation.
26. Curves and surfaces of degree 2.
27. Exercises. HA6.
28. Signature of a quadratic form. Sylvester criteria. Leading minors criteria.
29. Applications of operator technique: homogeneous linear differential equations with constant coefficients
30. Exercises.
31. Reserved.
32. Examination test.

Control tasks