Math 2406A: Abstract Vector Spaces Welcome to Math 2406A: Abstract Vector Spaces with Prof. Heitsch!
 Times & Places Lectures are in Skiles 243 on Mondays and Wednesdays from 3:05 -- 4:25. Office hours are currently scheduled for Monday and Wednesday from 4:30 -- 5:30pm in Skiles 226. (Meeting at other times is possible by appointment.) Announcements Handouts from Monday, August 17th: Course Syllabus and The Secrets of Student Success . Exam 1 is now scheduled for Wednesday, September 16th. Exam 1 will be given in Skiles 202, not Skiles 243. Alternate grading scheme: 0% Midterm 1, 25% Midterm 2, 25% Midterm 3, 40% Final Exam. Exam 2 is now scheduled for Wednesday, October 14th. Exam 2 will be given in Skiles 202, not Skiles 243. Exam 2 will cover up to and including Section 4.16. New material since Exam 1 begins with Section 3.14. Exam 3 is now scheduled for Wednesday, November 18th. Exam 3 will cover up to and including Section 5.22. New material since Exam 2 begins with Section 4.17. The final exam is scheduled for Monday, December 7th from 2:50 - 5:40pm in Skiles 202. The final exam will cover Chapters 1, 3, 4, 5, 6. You may bring one (1) 8.5 x 11 sheet of paper, written on both sides if you wish. There will be extra OH on Friday, Dec. 4th from 4:30 - 5:30pm. Homework Assignment 0 due Monday, August 24th. Assignment 1 due Monday, August 31st. Assignment 2 due Wednesday, September 9th. Assignment 3 due Wednesday, September 30th. Assignment 3 WITH HINTS due Wednesday, September 30th. Assignment 4 due Wednesday, October 7th. Assignment 5 due Wednesday, October 28th. Assignment 6 due Wednesday, November 4th. Assignment 7 due Monday, November 30th. Practice Problems Section 1.4 #5, 6, 7, 11. Section 1.8 #2, 5, 6, 7, 16, 17. Prove or disprove: the dot product is associative. Section 1.11 #8, 9, 14, 18. Section 1.15 #4, 6, 8, 9, 14, 17, 18. Truth tables for conditional, contrapositive, converse, inverse. What is a statement logically equivalent to the negation of the conditional? Justify your answer. Section 3.5 #9 - 20. As stated, one of the axioms for a linear space is redundant. Which one and why? Section 3.10 #11 - 22. Read the proofs of Theorems 3.5 and 3.7, and compare them with the proof of Theorem 1.10. Section 3.13 #5, 6, 8, 12, 14. Section 4.4 #11 - 15, 25 - 28. Let T: V -> W be a linear transformation from V into W, both linear spaces. Prove that the null space of T is a subspace of V. Let V be a Euclidean space and S a finite dimensional subspace of V with an orthonormal basis. Let T:V -> S be the linear transformation where T(f) is the projection of the element f from V onto the subspace S. Prove that N(T) = S^{perp}, the orthogonal complement of S. Section 4.8 #1, 2, 27, 28, 29, 30. Let V and W be linear spaces. Prove that the set of all linear transformations of V into W is a linear space. Let T: V -> W be a function from set V into set W. Prove that T has a left inverse if and only if T is one-to-one on V. Be very careful to correctly use the definition of left inverse, etc. Section 4.12 #4, 5, 8, 9, 19, 20. Section 4.16 #5, 8, 11, 12, 14, 15. Section 4.20 #8, 9, 10, 11, 12. Section 4.21 #7, 8, 9. Section 5.8 #2, 4, 7. Section 5.15 #1, 3, 5, 7. Section 5.20 #1, 2, 5. Section 5.20 #3, 4. Section 5.22 #2. Section 6.4 #1, 2, 4, 5, 6. Section 6.10 #5, 7, 9, 10, 14. Section 6.12 #4, 5, 6. Section 6.14 #1 - 7. Section 6.16 #3, 4, 9, 10, 11, 12. Things you should know Basic set theory and propositional logic. Here's a An Elementary Introduction to Logic and Set Theory. I have also just requested that the books Discrete Mathematics by Johnsonbaugh and _Discrete Mathematics and Its Applications_ by Rosen be put on reserve at the library. The first chapters will cover logic, sets, and basic proof techniques. Links Check your grades on T-Square. Student are expected to follow the Georgia Tech Academic Honor Code and Student Code of Conduct.
C. E. Heitsch
Fall semester 2009