# 3144/14001 Linear Algebra I - Spring 2009

## Syllabus

Instructor:
| Henning S. Mortveit | Email:
| henning@vt.edu |

Office I:
| 1111 RBXV, Corporate Research Center | Phone I:
| (231-5327) |

Office II:
| 419 McBryde Hall | Phone II:
| - |

Class hours:
| TR 9:30-10:45AM | Room:
| 222 Randolph Hall |

Prerequisite:
| Math 3034 or Math 3434 | ||

Office hours:
| TR 11:00AM-12:00PM in Office II - Also by appointment in Office I/II |

**Text/Syllabus:**Stephen H. Friedberg, Arnold J. Insel and Lawrence E. Spence:

*Linear Algebra*, 4

^{th}edition, Prentice Hall, 2003.

**Course goals:**From H. Dym's book Linear Algebra in Action:

"Linear algebra permeates mathematics, perhaps more so than any other single subject. It plays an essential role in pure and applied mathematics, statistics, computer science, and many aspects of physics and engineering."The goal of the course is for the student to acquire a working knowledge of the basic concepts of linear algebra. This includes vector spaces, linear transformations, matrix representations of a linear transformation after a choice of bases, elementary matrix operations, systems of linear equations, determinants, and diagonalization. We will cover most of the material in chapters 1 through 5, with selected material from chapter 7 as time permits.

**Exams:**There will be two in-class exams tentatively scheduled for Tuesday February 24 and Tuesday April 07. The two-hour final exam (Section 09T) is on May 12 from 10:05AM to 12:05PM. The final exam will take place in Randolph 222 unless stated otherwise. If you cannot take an exam at the scheduled time, please let me know as soon as possible and

*before*the exam. A make-up exam will be given for reasons that in my judgment are acceptable.

**Homework:**The course has 12 assignments. Generally assignments will be announced each Thursday, and will be due in class the following Tuesday. The exact schedule can be found here. It is subject to change - changes will also be announced in class. Late homework will only be accepted if handed in the first class following the due date, but only for half the credit.

*Very important:*The assignments are an integral part of this course. The 12 assignments should be considered a minimal effort, and working through additional problems is strongly encouraged.

**Attendance:**Will be taken, and will be kept for Mathematics Department records. Attendance may be used to adjust the final grade.

**Grading:**Is on a curve. However, 90% will be at least an A-, 80% will be at least a B-,70% will be at least a C-, and 60% will be at least a D-. Each assignment is worth 10 points, each in-class exam 40 points, and the final exam 50 points.

**Honor system:**The University Honor System is in effect for assignments and exams (see http://www.honorsystem.vt.edu). Discussion of class topics among students is encouraged, but the solutions to assignments that you hand in must be your own. All exams are closed-book, closed-notes.

**Students with special needs:**Students with disabilities, special needs or special circumstances should meet with the instructor during the first week of classes to discuss accommodations.

**General Notes:**Falling behind in this course is

*dangerous*, so turn in assignments on time, come to class prepared and take advantage of the office hours. (Also: read Professor Bud Brown's hints for success - available at http://www.math.vt.edu/people/brown/hints.html.)

**Supplementary literature:**

- H. Dym,
*Linear Algebra in Action*, AMS. - S. Axler,
*Linear Algebra Done Right*, Springer.

## Exams

Solution notes and comments will be posted here.### In-class exam 1: Tuesday February 24

A PDF copy of the exam with answers can be found here. (Password on syllabus.)### In-class exam 2: Tuesday, April 07

A PDF copy of the exam with answers can be found here. (Password on syllabus.)### Final exam: Tuesday, May 12

## Assignments

Assignments, solution notes and comments will be posted here.### Assignment 1: (Due Tuesday, January 27)

**Problems:**

- Section 1.2: 2, 7, 8, 12, 15, 18, 19.

**Notes:**Problems 12, 18 and 19 are worth 2 points each, the remaining problems are worth 1 point each.

**Solution:**Homework 1

### Assignment 2: (Due Tuesday, February 03)

**Problems:**

- Section 1.3: 2, 3, 5, 6, 8, 9, 10, 13, 22
- Bonus problems: Section 1.3: 17, 19.

**Notes:**Problem 8 is worth 2 points; all remaining problems are worth 1 point each. You may use the result of problem 4 without proof in problem 5. Also, whenever a problem is stated for a general field you may simply consider the real or complex numbers instead.

**Solution:**Homework 2

### Assignment 3: (Due Tuesday, February 10)

**Problems:**

- Section 1.4: 2d, 3abc, 4ab, 5hf, 6, 16
- Section 1.5: 2c, 5, 9, 10.
- Bonus problems: 1.4: 15 and 1.5: 19.

**Notes:**Each problem is worth 1 point. You should be able to complete all problems based on what we covered in class on Thursday February 5. For problem 1.5: 5 it may be helpful to look at example 4 of that section - it shows a similar case.

**Solution:**Homework 3

### Assignment 4: (Due Tuesday, February 17)

**Problems:**

- Section 1.6: 2b, 3b, 6, 7, 11, 13, 14, 17
- Bonus: 1.6: 20

**Notes:**Problems 14 and 17 are worth 2 points each. All remaining problems, including the bonus problem, are worth 1 point each.

**Solution:**Homework 4

### Assignment 5: (Due Thursday, March 5)

**Problems:**

- Section 2.1: 2, 5, 9cd, 10, 12, 20
- Section 2.2: 3, 4
- Bonus: 2.1: 18, 19

**Notes:**Each of the regular problems is worth 1.25 points (for the usual total of 10 points) while the bonus problems are worth 1 point each. On problems 9c and 9d it is sufficient to come up with a single example where linearity fails. For bonus problem 18 you may want to start by using the dimension theorem to determine dim(N(T)) and dim(R(T)).

**Solution:**Homework 5

### Assignment 6: (Due Wednesday, March 24)

**Problems:**

- Section 2.3: 2, 3, 9, 11
- Problem: Let T: P
_{2}(R) → P_{3}(R) be given by T(f(x)) = 2f'(x) + ∫_{0}^{x}3f(t) dt with basis β = {1, 1+x, 1+x+x^{2}} for P_{2}(R) and basis γ = {1, x, x^{2}, x^{3}} for P_{3}(R). Compute [T]_{β}^{γ}and use Theorem 2.14 to compute [T(1-x+x^{2})]_{γ}.

**Notes:**Each of the problems is worth 2 points.

**Solution:**Homework 6

### Assignment 7: (Due Tuesday, March 31)

**Problems:**

- Section 2.4: 2cf, 3, 14
- Section 2.5: 2d, 4, 5, 10
- Bonus 2.4: 17, 20

**Notes:**The regular problems will count for the usual 10 points (equal weight) whereas the bonus problems are worth 1 point each. Note that for problem 10 in section 2.5 you can use the result of problem 13 in section 2.3 without proof, but you are of course welcome to include it. Note that bonus problem 20 uses the result of bonus problem 17. Also, 17a was a part of an earlier homework - you can omit part a) if you want to.

**Solution:**Homework 7

### Assignment 8: (Due Tuesday, April 14)

**Problems:**

- Section 3.2: 2be, 4, 5bc, 7

**Notes:**Each problem is worth 2.5 points. On problem 5 b & c you only need to compute the rank.

**Solution:**Homework 8

### Assignment 9: (Due Tuesday, April 21)

**Problems:**

- Section 3.2: 5d, 6ad, 15.
- Section 3.3: (2+3)ae, 7bc.

**Notes:**Problems 2 and 3 from Section 3.3 should be regarded as one problem. Solve 2a and 3a together and solve 2e and 3e together.

**Solution:**Homework 9

### Assignment 10: (Due Thursday, April 23)

**Problems:**

- Section 3.4: 2ag, 7, 10, 12

**Solution:**Homework 10

### Assignment 11: (Due Tuesday, April 28)

**Problems:**

- Section 4.1: 2, 5, 6, 7, 8
- Section 4.2: 5, 17
- Section 4.3: 9, 11, 12

**Notes:**All problems are worth 1 point. You may use the formula of Problem 25 in Section 4.2 without proof, that is, det(kA) = k

^{n}det(A), where-ever it applies.

**Solution:**Homework 11

### Assignment 12: (Due Thursday, April 30)

**Problems:**

- Section 5.1: 2cd, 3ab, 4e, 9, 12
- Bonus 5.1: 20, 21

**Notes:**The regular problems are worth 2 points each; the bonus problems are worth 1 point each.

**Solution:**Homework 12

Thu Jan 8 16:24:47 EST 2009