Advanced Linear Algebra & Vector Analysis

Course Information

• Independent Study (ISP) Proposal - Form

• Syllabus - Advanced Linear Algebra and Vector Analysis

Very important: learning math is about doing problems. That is how you get better. While the complete answers to all questions are available to you, it is crucial that before consulting an answer, you must give the question a good try on your own. If you read the question and just immediately read the answer, then you will not learn. 

Advanced Linear Algebra

Logic and Proof Unit

Much of this material is also covered in the appendix of the Linear Algebra textbook, although it is covered in more depth here. 

• Assignments

• Logic and Proof - Reading and Exercises

• Set Operations - Reading and Exercises

• Relations - Reading and Exercises

• Functions - Reading and Exercises

• Mathematical Induction - Reading and Exercises

• Additional Reading and Exercises: Construction of Numbers

• Errata for Steven R. Lay's Analysis with an Introduction to Proof (4th Edition)

• Solutions

Logic and Proof Lecture Notes

• Logic

• Techniques of Proof

• Set Operations

• Relations

• Functions

• Natural Numbers and Mathematical Induction

Handouts

• Transformations

• Geometry of Eigenvectors

Lecture Summaries

Please note: the posted lecture notes were originally intended for the instructor's purposes only and may not be complete. By no means are the posted lecture notes meant to serve as a substitute for attendance in class and the reading of the text. It is not guaranteed that they are without error either. Should there be any errors, notify and inform the instructor of the error immediately or as soon as possible. 

Chapter 1: Linear Systems

1. Solving Linear Systems

2. Matrices and Vectors

3. Reduced Echelon Form

Chapter 2: Vector Spaces

4. Vector Spaces

5. Spaces and their Subspaces

6. Linear Independence

7. Basis

8. Dimension

9. Vector Spaces and Linear Systems

Chapter 3: Maps Between Spaces

10. Isomorphisms

11. Dimension and Isomorphisms

12. Homomorphisms

13. Rank + Nullity

14. Range Space and Null Space

15. Computing Linear Maps

16. Matrices and Linear Maps

17. Computing Range and Null Spaces

18. Matrix Operations

19. Mechanics of Matrix Multiplication

20. Inverses

21. Change of Basis

Chapter 4: Determinants

22. Determinants

23. Laplace's Expansion

Chapter 5: Similarity

24. Complex Representations

25. Similarity

26. Diagonalizability

27. Eigenvalues and Eigenvectors

Multiple Integration - Lecture Summaries

• Double Integrals over Rectangles

• Iterated Integrals

• Double Integrals over General Regions

• Double Integrals in Polar Coordinates

• Applications of Double Integrals

• Triple Integrals

• Triple Integrals in Cylindrical Coordinates

• Triple Integrals in Spherical Coordinates

• Change of Variables

• Gaussian Integral

Vector Calculus - Lecture Summaries

• Vector Fields

• Line Integrals I

• Line Integrals II

• Conservative Vector Fields

• Curl and Divergence

• Parametric Surfaces and their Area

• Surface Integrals

• Green's Theorem

• Stokes' and Divergence Theorem

Worksheets

• Worksheet on Change of Variables & Solutions

• Worksheet on Line Integrals & Solutions

• Worksheet on Curl and Divergence & Solutions

• Worksheet on Surface Integrals and Flux & Solutions

• Worksheet on Integral Theorems & Solutions

ODes (if time allows)

• Applied ODEs - Lecture Notes 1

  • Ordinary Differential Equations - Assignment 1

  • Solutions

• Applied ODEs - Lecture Notes 2

  • Ordinary Differential Equations - Assignment 2

  • Solutions

• Applied ODEs - Lecture Notes 3

• Applied ODEs - Lecture Notes 4

• Applied ODEs - Lecture Notes 5

• Applied ODEs - Lecture Notes 6

  • Ordinary Differential Equations - Assignment 3

  • Solutions

  • Table of Laplace Transforms

• Applied ODEs - Lecture Notes 7

  • Ordinary Differential Equations - Assignment 4

    • Solutions

  • Ordinary Differential Equations - Assignment 5

    • Solutions

• Applied ODEs - Lecture Notes 8

• Applied ODEs - Lecture Notes 9

• Additional Notes