MA552 Axiomatic Linear Algebra
Instructor: Nikolay S. Strigul
E-mails: nstrigul@stevens.edu, nstrigul@princeton.edu
Office Hours: by appointment
Lectures: Tuesdays 06:15-08:45 pm, Babbio Center, 104
Homework: There will be weekly homework assignments,
due every Tuesday, when the correct solutions will be given out.
Exams, Quizzes: There will be Midterm and Final Exams.
Every class will start with a closed book quiz consisting of definitions, theorems, and some past homework problems.
Grading:
- Quizzes: 35 %
- Homework assignments: 25 %
- Midterm: 10 %
- Final: 30 %
General comments:
MA552 is a graduate-level course in linear algebra. The major goal of this
course is to consider, from the general axiomatic point of view, basic ideas
of finite dimensional vector spaces. Also some advanced
topics—such as multilinear algebra, and normed, inner product, and topological
vector spaces—will be considered, provided that the mandatory material is covered.
It is assumed that students have already taken an undergraduate linear algebra course,
such as MA232. However, to assure that everyone is familiar with the undergraduate
material and to acclimate to the general axiomatic approach through intuitive examples,
this course will start with an extensive review of the elementary linear algebra topics:
geometrical vector theory, linear transformations in two- and three-dimensional Euclidean
spaces, real matrices, determinants, eigenvalues, and eigenvectors. This review will correspond
to the first 5 chapters of the Gilbert Strang's book and will also include a discussion
of important applications (in particular to systems of differential equations),
and should take up the first 4-5 lectures. The second part of the course will
be completely devoted to the formal axiomatic approach to finite-dimensional vector spaces.
Textbooks
1) Linear Algebra by K. Hoffman and R. Kunze
2) Finite-Dimensional Vector Spaces by P.R. Halmos
3) Linear Algebra and Its Applications by Gilbert Strang, 4th Edition, 2006
4) Linear Algebra and Multidimensional Geometry by N.V. Efimov and E.R. Rozendorn
5) Linear Algebra and Analysis by M. Zamansky
6) Introduction to Real Analysis by A.N. Kolmogorov and S.V. Fomin
Course program:
Aug 29. Lecture 1. - Linear Equations and Vector Spaces.
1. Introduction.
2. The geometry of linear equations.
3. Gaussian elimination.
4. Matrix notation.
5. Fields.
6. Definition of a vector space over the field F.
Sep 5. Lecture 2. - Review of Matrices. Linear Independence.
7. Matrix operations.
8. Inverses and transposes.
9. Special matrices and applications.
10. Linear independence.
11. The solution of m equations in n unknowns.
12. Basis and dimension.
Sep 12. Lecture 3. - Orthogonality.
13. Orthogonal vectors in the Euclidean spaces.
14. Definition of the inner product.
15. Norm of the vector.
16. Orthonormal basis.
17. Fourier series.
Sep 19. Lecture 4. - Determinants.
18. Even and odd permutations.
19. Signature of a permutation.
20. Definition of the determinant, the Leibniz formula.
21. Properties of the determinant.
22. Methods to compute the determinant.
Sep 26. Lecture 5. - Eigenvalues and Eigenvectors.
23. Matrix representation of linear transformations.
24. Eigenvectors and eigenvalues of linear transformations.
25. Eigenvalue equation.
26. Finding eigenvalues and eigenvectors.
27. Solving linear recurrence relations.
28. Solving systems of linear differential equations.
Oct 3. Lecture 6. - A General Approach to Vector Spaces.
29. Internal composition laws.
30. Groups, rings, fields.
31. External composition laws. Vector spaces.
32. Construction of a vector space over a field.
33. Vector subspaces, linearly independent elements, basis.
34. Basic properties of finite-dimensional vector spaces.
Oct 10. No lecture (Stevens is on Monday schedule ) - takehome midterm exam.
Oct 17. Lecture 7. - Linear Transformations.
35. Linear transformation, linear form and endomorphism.
36. Operations on linear mappings.
37. Properties of linear mappings.
38. Isomorphism.
39. Linear mappings in vector spaces of finite dimension.
Oct 24. Lecture 8. - Rank of a linear mapping.
40. Direct sum.
41. Quotient space.
42. Rank of a linear mapping.
43. Linear forms.
Oct 31. Lecture 9. - Transpose of a linear mapping.
44. Dual spaces.
45. Linear forms.
46. Linear equations in finite-dimensional vector spaces.
Nov 7. Lecture 10. - Convex Functionals. The Hahn-Banach theorem.
47. Convex sets and bodies.
48. Convex functionals.
49. The Minkowski functional.
50. The Hahn-Banach theorem.
Nov 14. Lecture 11. - Normed Linear Spaces.
51. Definitions of norm and normed linear space. Examples.
52. Subspaces of a normed linear space.
53. Metric induced by a norm.
54. Euclidean Spaces. Scalar product and orthogonality.
55. Norm induced by a scalar product.
Nov 21. Lecture 12. - Multilinear Algebra.
56. Bilinear mappings.
Nov 28. Lecture 13. - Tensors.
56. The tensor product of two spaces.
57. Mulitiliear mapping and tensor products.
Dec 5. Lecture 14. - Exterior Product.
58. Exterior power of order 2.
59. Definition of an alternating multilinear mapping.
60. Definition of the exterior power.
61. Determinants.
Dec 12. Final exam.
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