If you are a student of Math 2568 in Spring 2017, go to The course webpage for Spring 2017 .

**Some of the links were deleted/disabled at the end of the semester.**

Contents

- References
- Homework assignments
- HW 1 Due 9/2 in class
- HW 2 Due 9/9 in class
- HW 3 Due 9/16 in class
- HW 4 Due 9/23 in class
- HW 5 Due 9/30 in class
- HW 6 Due 10/7 in class
- HW 7 Due 10/12 Wednesday in class
- HW 8 Due 10/21 in class
- HW 9 Due 10/28 in class
- HW 10 Due 11/4 in class
- HW 11 Due 11/9 Wednesday in class
- HW 12 Due 11/18 in class
- HW 13 and HW 14 combined Due 11/30 Wednesday in class

- Extra exercise problems
- Q&A for Math 2568

We meet:

- MWF 13:50-14:45
- Mendenhall

Lab 191

11/30 Practice Problems for Final exam were posted.

11/16 HW 13 and HW 14 were combined and due 11/30 Wednesday.

10/28 No office hour today (10/28 3PM).

10/23 Solutions for midterm 2 practice problems were posted.

10/11 Practice problems for midterm exam 2 were added.

10/9 The extra problems page was added.

9/15 No office hour on Friday (9/16 3pm).

9/14 Answer keys for practice problems for midterm 1 was posted.

9/7 Practice problems for midterm 1 were added.

8/31 Q&A site was added.

8/29 The grader information is added. See the syllabus.

8/26

- PDF version of Preview Lecture notes are also available in Evernote or Dropbox.
- Pictures of problems for the first homework were added.

8/24 First class.

**Final exam results**

Maximal score 200 (3 students)

Next best score 198 (2 students)

Minimal score 86

Average of all students:151/200 (75.6%)

Average of students with absence less than 4 times: 156 (78%)

Average of students with absence more than 3 times: 126 (63%)

The main topics we will cover in 2568 include Matrix algebra, vector spaces and linear maps, bases and dimension, eigenvalues and eigenvectors, applications.

### Lecture Notes

(Lecture 1-29) |
(Lecture 30-) |

You can find

- Lecture notes for preview
- Lecture notes used in the lecture

at OneDrive.

You may create a OneDrive account here, and you and I both get extra 0.5 GB free storage.

### Lecture Notes in PDF

Here you can find lecture notes in PDF file in Evernote or Dropbox.

Sometimes, the conversion to PDF does not work properly.

If you see a broken PDF, please use OneNote version.

You may create a Evernoet account here or create a Dropboxt here.

### Textbook

The required textbook is

Introduction to Linear Algebra, 5th edition

by L.W. Johnson, R.D. Riess, and J.T. Arnold, published by Pearson,

ISBN Softcover: 0321628217, Hardcover: 0201658593

I will assign homework problems mostly from the textbook. So please buy/rent the textbook as soon as possible.

## References

Some past exam problems and solutions of 2568 linear algebra at the Ohio State University.

These are linear algebra exam problems from various universities.

Solutions are also available.

Problems/solutions in linear algebra by topics.

You should be able to solve most of these problem after taking 2568 linear algebra course.

Several random problems in linear algebra.

## Homework assignments

### HW 1 Due 9/2 in class

- Day 1 §1.1 p.12- #2, 20, 28.

Also take a quiz here. (No need to submit this quiz) - Day 2 §1.2 p. 26- # 18, 30, 44

For those who has not received the textbook yet, pictures of the problems are available here (only this time).

Comment from the grader: "many students confused HW problem #18 with former question of the section 1.2."

### HW 2 Due 9/9 in class

- Day 3 §1.3 p.37- # 4, 12, 23

Also take a quiz here. (No need to submit this quiz) - Day 4 §1.5 p.58- # 48, 58, 68
- Day 5 §1.6 p.69- # 30, 43, 46

Comment from the grader: For problem 58, many students didn't recognize it and some students gave the only answers without justifying it.

Most students do a good job.

### HW 3 Due 9/16 in class

- Day 6 §1.7 p.78- # 30, 49, 50

Also take a quiz here. - Day 7 §1.7 p.78- # 47, 51, 52

Comment from the grader: some students don't know how to write the answer properly for problem #50 and #51.

### HW 4 Due 9/23 in class

- Day 8 §1.9 p.102- # 7, 19, 48
- Day 9 Any three problems from practice problems for midterm 1. You need to write down your proof/explanation even if the problems you chose are from true and false, or multiple choice.

Comment from the grader: Most students did the homework well. Even though, some students didn't look at the problem carefully, especially #19.

A few students seem to copy other's soluions or just writed down the answer without reasoning.

### HW 5 Due 9/30 in class

- Day 11 §3.2 p.174- # 16, 18, 30
- Day 12 §3.2 p.174- # 21, 27, 31

For #18 and #21 see this post and imitate the proof. But do not just copy.

Here is the proof that the null space of a matrix is a subspace.

- Day 13 §3.3 p.186- # 17, 34, 46

Comment from the grader: I graded three problems and each problem is 3 points. The total is 9 points and I entered 9+1 points on Carmen. Many students did the homework well.

### HW 6 Due 10/7 in class

- Day 14 §3.3 p.186- # 30, 32, 40 iii & iv
- Day 15 §3.4 p.200- # 25, 27, 31

Here is my solution of problem 25 b). Please use it as a guide for how to write a solution like this problem.

- Day 16 §3.4 p.200- # 12, 18, 24

Comment from the grader:There are some students who did not solve a few problems. I want you to give the message that even though the problem is hard and you don't know how to solve, please give your draft idea or some note for the solution. Then I will refer it for grading.

Some students are confused about the leading 1's method and finding the Null space with Range space.

### HW 7 Due 10/12 **Wednesday** in class

Note that the due date is on Wednesday 10/12.

- Day 17 §3.5 p.212- # 22, 26, 27
- Day 18 §3.5 p.212- # 24, 28, 31
- Day 19 no problems for today

Comment from the grader: Most students become skillful at finding null space and range space.

### HW 8 Due 10/21 in class

- Day 20 §5.2 p.366- # 21, 22, 23
- Day 21 §5.3 p. 373- # 19, 24, 31

Here is a problem similar to homework problems.

Comment from the grader: I graded section 5.2 # 21, section 5.3 #19, #31. If they don't solve one of them or more with nothing, they get no point. As a result of grading, some students didn't solve one or more of them. Since I just pick problems for grading randomly or with some reasons, I want to ask them to solve all problems not partial. Most of them are pretty good at problems and I want to recommend them not to copy friend's homework without understanding.

### HW 9 Due 10/28 in class

- Day 22 §5.4 p.386- # 7, 10, 23
- Day 23 §5.4 p.386- # 22, 24, 26
- Day 24 §3.7 p.239- # 24, 26, 30

Comment from the grader: I graded #10, #22 from section 5.4 and #26 from section 3.7.

Some students need to go over the definition of symmetric matrix.

Most students do the homework well.

### HW 10 Due 11/4 in class

- Day 25 §3.7 p.239- 12, 20, 22

Review midterm 2 exam problems. Especially, these writing problems

Comment from the grader: I graded all of them, 3 questions.

For the first question, I think some students misundersand the equation (6) from section 3.7. on the textbook about forming the matrix of linear transformation T.

And...most students do the homework well.

### HW 11 Due 11/9 **Wednesday** in class

- Day 28 §3.6 p.224- # 12, 16, 19
- Day 29 §4.1 p.279- # 10, 13, 17
- Day 30 No Homework problem today

Comment from the grader: I graded #16, #19 from 3.6 and#13 from 4.1. Most students follow the Gram-Schmidt process well. But, some of them made a mistake during calculation for the process where they lost their points.

### HW 12 Due 11/18 in class

- Day 31 §4.2 p.288- # 19, 23, 24
- Day 32 §4.4 p.305- # 4, 8, 11

Comment from the grader: I graded #19, #24 from section 4.2 and #11 from section 4.4.

Most students did a good job doing their homework.

### HW 13 and HW 14 combined Due 11/30 **Wednesday** in class

(This is the last homework.)

- Day 33 §4.4 p.305- # 18, 19, 21

(I changed problem #22 to #21.) - Day 34 §4.5 p.314- # 6, 14, 17
- Day 35 §4.5 p.314- # 16, 21, 26
- Do and submit the problems in the following PDF file:
- Day 36 §4.6 p.324- # 16, 19, 20

Comment from the grader: I graded #18, 21 from section 4.4, #14, 16, 26 from section 4.5, #16, 20 from section 4.6 and one of the pdf file problems.

Most students got full score if they did solve problems. I marked down some comments and total point per each problem on their homework.

## Extra exercise problems

Please do not submit these extra problems with homework assignments.

### Extra 1

- Day 1 § 1.1. p.12- #1, 4, 8, 11, 24, 25
- Day 2 § 1.2. p.26- # 9, 20, 24, 28, 37, 54

### Extra 2

- Day 3 § 1.3. p.37- #1, 8, 10, 14, 17, 20, 22, 24
- Day 4 $ 1.5. p.57- #1, 2, 3, 6, 8, 12, 25, 26, 30, 31, 32, 36, 38, 42, 53, 54, 55, 56, 65, 66
- Day 5 $1.6. p.69- # 4, 5, 7, 8, 10, 12, 14, 18, 20, 22, 25, 26, 29, 35, 50, 56, 57

### Extra 3

- Day 6 §1.7 p.78- # 1, 2, 6, 14, 16, 22, 23, 31, 32, 34, 35, 39, 40, 46, 48, 51, 52, 53

### Extra 4

- Day 8 §1.9 p.102- # 1, 8, 9, 13, 18, 21, 24, 27, 28, 31, 67

### Extra 5

- Day 11, 12 §3.2 p.174- # 1, 2, 3, 6, 7, 10, 11, 13, 19, 31
- Day 13 §3.3 p.186- # 1, 4, 9, 14, 20, 26, 30, 32, 38, 40, 42, 43, 49

### Extra 6

- Day 15 §3.4 p.200- # 6, 7, 12(b), 14(b), 32, 33
- Day 16 §3.4 p.200- # 11, 13, 14, 17, 23, 27

### Extra 7

- Day 17 §3.5 p.212- # 4, 8, 12, 18, 29, 31
- Day 19. Here is a proof of the fact that the rank of a matrix is equal to the rank of the transpose matrix.

### Extra 8

- Day 21 §5.3 p. 373- # 19, 24, 31

### Extra 9

- Day 22 §5.4 p.386- # 1, 5, 11, 14, 36, 37
- Day 23 §5.4 p.386- # 18, 25, 27, 29, 30, 34, 35
- Day 24 §3.7 p.239- # 3, 4, 10, 14, 19, 23, 29, 38, 40

For problem #29, 30 use the matrix $A_3=\begin{bmatrix}

0 & 0\\

2 & 0

\end{bmatrix}$

### Extra 11

- Day 28 §3.6 p.224- # 1, 7, 15, 18, 22, 28
- Day 29 §4.1 p.224- # 5, 6, 7, 18

### Extra 12

- Day 31 §4.2 21, 26, 29, 33, 34
- Day 32 §4.4 p.305- # 3, 7, 9, 12, 14

### Extra 13&14

- Day 33 §4.4 p.305- # 12, 17, 24, 25, 27
- Day 34 §4.5 p.314- # 4, 10, 11, 13
- Day 35 §4.5 p.314- # 22, 23, 25
- Day 36 §4.6 p.324- # 8, 10, 21, 29, 36-42

### Extra

## Q&A for Math 2568

The Q&A broad for students taking Math 2568 at OSU.

You can ask and answer questions regarding Math 2568 linear algebra.

Help each other.