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 %

Course programCourse program (PDF)

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.

Handout.pdf Homework 1.pdf

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.

Homework 2.pdf Homework 1 solutions.pdf Quiz 1.pdf

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.

Homework 3.pdf Homework 2 solutions.pdf Quiz 2.pdf

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.

Homework 4.pdf Homework 3 solutions.pdf Quiz 3.pdf

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.

Homework 5.pdf Homework 4 solutions.pdf Quiz 4.pdf

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.

Homework 6.pdf Homework 5 solutions.pdf Quiz 5.pdf

Oct 10. No lecture (Stevens is on Monday schedule ) - takehome midterm exam.

Midterm exam.pdf Midterm exam solutions.pdf 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.

Homework 7.pdf Homework 6 solutions.pdf Quiz 6.pdf

Oct 24. Lecture 8.- Rank of a linear mapping.

40. Direct sum.

41. Quotient space.

42. Rank of a linear mapping.

43. Linear forms.

Homework 8.pdf Homework 7 solutions.pdf Quiz 7.pdf

Oct 31. Lecture 9.- Transpose of a linear mapping.

44. Dual spaces.

45. Linear forms.

46. Linear equations in finite-dimensional vector spaces.

Homework 9.pdf Homework 8 solutions.pdf Quiz 8.pdf

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.

Homework 10.pdf Homework 9 solutions.pdf Quiz 9.pdf

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.

Homework 11.pdf Homework 10 solutions.pdf Quiz 10.pdf

Nov 21. Lecture 12.- Multilinear Algebra.

56. Bilinear mappings.

Homework 12.pdf Homework 11 solutions.pdf Quiz 11.pdf

Nov 28. Lecture 13.- Tensors.

56. The tensor product of two spaces.

57. Mulitiliear mapping and tensor products.

Homework 13.pdf Homework 12 solutions.pdf Quiz 12.pdf

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.

Homework 13 solutions.pdf Quiz 13.pdf

Dec 12. Final exam.

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