MATH 510 Linear Algebra

Spring 2024

Instructor

Grader

Time and Place

Texts and Resources

Course Content

Advanced topics in linear algebra including canonical forms; unitary, normal, Hermitian and positive-definite matrices; variational characterizations of eigenvalues.

The course is roughly divided into three parts:

Part 1 focuses on algebraic aspects, and we cover [HK, Ch. 1–7].

Part 2 concerns geometric aspects, from [HK, Ch. 8–9] and [HJ, Ch. 2, 4–5].

Part 3 will be on analytic properties, from [HJ, Ch. 4, 6– 7].

Exams and Grading

Weekly homework, due most Fridays.

Homework

Homework sets will be posted here. They are due most Fridays. You are strongly encouraged to use LaTeX to typeset your homework solutions. You may download and use the homework LaTeX file as a starting point. Overleaf is a popular online LaTeX environment which offers helpful tutorials.

  1. (Due Jan 26) Homework 1, (LaTeX file)
  2. (Due Feb 2) Homework 2, (LaTeX file)
  3. (Due Feb 9) Homework 3, (LaTeX file)
  4. (Due Feb 16) Homework 4, (LaTeX file)
  5. (Due March 8) Homework 5, (LaTeX file)

Lecture Summary

A short summary of each lecture will be posted here.

HK = Hoffman and Kunze, HJ = Horn and Johnson, H = Hartwig

  1. Fields [HK §1.1], vector spaces [HK §2.1], linear independence [HK §2.3]
  2. Bases and dimension [HK §2.3], subspaces and sums of subspaces [HK §2.2]
  3. Matrices, transpose and hermitian adjoint
  4. Linear maps, coordinates of vectors, matrices of linear transformations, [HK §2.4 and §3.1–§3.4]
  5. Change of basis [HK §3.4], (External) direct sums of any vector spaces
  6. Quotient spaces [HK Appendix A.4]
  7. Tensor products (Lecture notes)
  8. Modules over \(F[x]\) (Notes on polynomials)
  9. Minimal polynomial Lecture notes, [HK §6.3], [HJ §3.3]
  10. Primary Decomposition Theorem, [HK §6.8]
  11. Abstract Jordan decomposition, [HK §6.8, Thm. 13], Jordan normal form (special case), [HK §7.3]
  12. Normal Form for Nilpotent Linear Maps and Jordan’s Normal Form, see [H §5 in v0.2], [HK §7.3]
  13. Evaluating polynomials, Real Jordan Form, [HJ §3.4.1], determinants, [H §5.3-4, §6 in v0.2]
  14. Invariant Factors I (Lecture notes), [HK §7.4]
  15. Invariant Factors II (Lecture notes), [HK §7.4]
  16. (Review)
  17. (Review)
  18. Inner products, norms, Cauchy-Schwarz and triangle inequalities [HJ §5.1]; unitary matrices [HJ §2.1]
  19. More on unitary matrices; \(U(n)\) is a compact topological group [HJ §2.1]
  20. Matrices \(A\) such that \(A^\ast\sim A^{-1}\) [HJ §2.1]; Schur’s Unitary Triangularization Theorem [HJ §2.3]; some consequences [HJ §2.4]

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