# 2568 Linear Algebra Spring 2017

### Course Information

Some links were disabled after the semester.

Instructor: Yu Tsumura

Where: Stillman Hall 235

When: MWF 12:40-1:35

The syllabus for Math 2568.

### News

News
4/6 Midterm 2 problems and solutions were posted.
2/25 Office hour on Monday Feb. 27th is 10:30-11:30
2/13 Midterm Exam 1 solutions were posted.
2/1 Midterm Exam 1 info was added.
1/9 First Class

### Textbook

The required textbook is

Introduction to Linear Algebra, 5th edition,
ISBN Softcover: 0321628217, Hardcover: 0201658593

## The Next Quiz and Homework

Quizzes will be based on the following homework problems.

## Final Exam

The Final exam is scheduled for May 2nd 12:00-1:45 PM (Tuesday) in the usual class room.
The exam will cover the materials that we studied through the semester.

Please review lecture notes, homework problems, quiz problems.

### List of Quiz Problems of Linear Algebra

There were 13 weekly quizzes. Here is the list of links to the quiz problems and solutions.

## Midterm 2

Midterm Exam 2 is scheduled for March 31st (Friday) in class.
The exam will cover the materials that we studied in Chapter 3 and 5 of the textbook.

Please review lecture notes, homework problems, quiz problems.

Errata: In the handout, (2.2) is False and (2.3) is True. In (2.6), it should be "the zero vector of $\R^m$ to the zero vector to $\R^n$."
In (3.6), it should be "sends the zero vector of $\R^m$ to the zero vector of $\R^n$."

## Midterm 1

Midterm Exam 1 is scheduled for February 10th (Friday) in class.
The exam will cover the materials we studied in Chapter 1 of the textbook.
(Section 1.4 and 1.8 are excluded.)

Please review lecture notes, homework problems, quiz problems.
There are supplementary/conceptual exercises in the textbook starting on page 105.

### Midterm 1 problems and solutions

Max 80/80 Min 10/80. The class average is 61 (76%).
The problems and solutions.

1. Problem 1 and its solution: Possibilities for the solution set of a system of linear equations
2. Problem 2 and its solution: The vector form of the general solution of a system
3. Problem 3 and its solution: Matrix operations (transpose and inverse matrices)
4. Problem 4 and its solution: Linear combination
5. Problem 5 and its solution: Inverse matrix
6. Problem 6 and its solution: Nonsingular matrix satisfying a relation
7. Problem 7 and its solution: Solve a system by the inverse matrix
8. Problem 8 and its solution: A proof problem about nonsingular matrix

### Where these problems came from

Problem 1, 2, 5 are typical.
Problem 3 tests whether you understand matrix operation including transpose and inverse matrices.
(Review Theorem 10 (p.64) and Theorem 17 (p.99).)
Problem 4 is exactly taken from lecture notes 6. Problem 6 is the same problem as Quiz 4.
Problem 7 is a combination of the homework problem #31 in section 1.9 and #43 in section 1.6.
The idea of Problem 8 is the same as the homework problem #52 in section 1.7.
(No need to know this but the matrix of Problem 5 is the transpose of the matrix in Quiz 4.)

## Lecture Notes

1. §1.1 Introduction to Matrices and Systems of Linear Equations
2. §1.2 Echelon Form and Gauss-Jordan Elimination
3. §1.3 Consistent Systems of Linear Equations
4. §1.5 Matrix Operations
5. §1.6 Algebraic Properties of Matrix Operations
6. §1.7 Linear Independence and Nonsingular Matrices
7. §1.7 Linear Independence and Nonsingular Matrices Part 2
8. §1.7 Linear Independence and Nonsingular Matrices Part 3
9. §1.9 Matrix Inverse and Their Properties
10. §1.9 Matrix Inverse and Their Properties Part 2
11. §3.2 Vector Space Properties of $\R^n$
12. §3.3 Examples of Subspaces
13. §3.3 Examples of Subspaces Part 2
14. Midterm 1
15. §3.3 Examples of Subspaces Part 3
16. Review of Midterm 1
17. §3.4 Bases for Subspaces
18. §3.4 Bases for Subspaces Part 2
19. §3.5 Dimensions
20. §3.5 Dimensions Part 2
21. §5.2 Vector Spaces
22. §5.3 Subspaces
23. §5.3 Subspaces Part 2
24. §5.4 Linear Independence, Bases, and Coordinates
25. < §5.4 Linear Independence, Bases, and Coordinates Part 2
26. §5.4 part 3 & §3.6 Orthogonal Bases for Subspaces
27. §3.6 Orthogonal Bases for Subspaces Part 2 & §3.7 Linear Transformation from $\R^n$ to $\R^m$
28. §3.7 Linear Transformation from $\R^n$ to $\R^m$. Part 2
29. §4.1 The Eigenvalue Problem for $(2\times 2)$ Matrices
30. §4.2 Determinants and the Eigenvalue Problem
31. §4.2 Determinants and the Eigenvalue Problem Pat 2
32. Exam 2.
33. §4.4 Eigenvalues and the Characteristic Polynomial
34. §4.5 Eigenvectors and Eigenspaces
35. §4.5 Eigenvectors and Eigenspaces Part 2
36. §4.6 Complex Eigenvalues and Eigenvectors
37. §4.7 Similarity Transformation and Diagonalization
38. §4.7 Similarity Transformation and Diagonalization part 2
39. §4.7 Similarity Transformation and Diagonalization part 3
40. Practice problems for the final exam 1
41. Practice problems for the final exam 2
42. Practice problems for the final exam 3

## Upcoming quizzes

Homework problems are tentative and may change without a notice.

## Past quizzes

Our grader is Ibrahima Daw (his office MA 420).

1) LER: Logical error (-2 pts)
2) NES: Not enough steps shown (-1pt)
3) IS: Incomplete Solution (fraction of the total points of the problem)
4) CE: Computation Error (-1 point)
5) CAWM: Correct Answer wrong math (-2pts).

### Quiz 1 (Jan. 18th)

Here are Quiz 1 problems and solutions.

• §1.1 p.12- #2, 4, 8, 11, 20, 24, 25, 28.
• §1.2 p.26- #9, 18, 20, 24, 28, 30, 37, 44, 54
• §1.3 p.37- #1, 4, 8, 10, 12, 14, 17, 20, 22, 23, 24

Take the following practice quizzes
Practice quiz 1
Practice quiz 2
Practice quiz 3

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

### Quiz 2 (Jan 25th)

Here are Quiz 2 problems and solutions.

• §1.5 p.58- #1, 2, 3, 6, 8, 12, 25, 26, 30, 31, 32, 36, 38, 42, 48, 53, 54, 55, 56, 58, 65, 66, 68
• §1.6 p.69- #4, 5, 7, 8, 10, 12, 14, 18, 20, 22, 25, 26, 29,30, 35, 43, 46, 50, 56, 57

### Quiz 3 (Feb. 1st)

Here are Quiz 3 problems and solutions

§1.7 p.78- # 1, 2, 6, 14, 16, 22, 23, 30, 31, 32, 34, 35, 39, 40, 46, 47 48, 49, 50, 51, 52, 53

### Quiz 4 (Feb. 8th)

Here are Quiz 4 problems and solutions
§1.9 p.102- # 1, 7, 8, 9, 13, 18, 19, 21, 24, 27, 28, 31, 48, 50, 67

### Quiz 5 (Feb. 15th)

Here are Quiz 5 problems and solutions
§3.2 p.174- # 1, 2, 3, 6, 7, 10, 11, 13, 16, 18, 19, 21, 30, 31

### Quiz 6 (Feb. 22th)

Here are Quiz 6 problems and solutions.

§3.3 p.186- # 1, 4, 9, 14, 17, 20, 26, 30, 32, 34, 38, 40, 42, 43, 46, 49, 50
§3.4 p.200- # 1, 6, 9, 12 (a), (b), 25, 27, 30

### Quiz 7 (Mar. 1st)

Here are Quiz 7 problems and solutions.

§3.4 p.200- # 11, 12, 13, 14, 18, 24, 31, 32, 33, 36
§3.5 p.212- # 4, 8, 12, 18, 22, 24, 26, 27, 28, 29, 31

### Quiz 8 (Mar. 8th)

Here are Quiz 8 problems and solutions.

§5.2 p.366- # 8, 12, 15, 17, 21, 22, 23, 27
§5.3 p.373- # 4, 8, 10, 19, 22, 24, 27, 31

### Quiz 9 (Mar. 22nd)

Here are Quiz 9 problem and its solution.

§5.4 p.386- # 1, 5, 7, 10, 11, 14, 18, 22, 23, 24, 25, 26, 27, 29 (see below), 30, 34, 35, 36, 37

For problem #29, use $A_3=\begin{bmatrix} 0 & 0\\ 2& 0 \end{bmatrix}$.

### Quiz 10 (Mar. 29th)

Here are Quiz 10 problems and solutions.

§3.6 p.224- # 1, 7, 12, 15, 18, 19, 22, 28
§3.7 p.239- # 3, 4, 10, 12, 14, 19, 20, 22, 29, 38, 40

### Quiz 11 (Apr. 5th)

Here are Quiz 11 problems and solutions.
§4.1 p.279- # 5, 6, 7, 10, 13, 17, 18
§4.2 p.288- 19, 21, 23, 24 26, 29, 33, 34

### Quiz 12 (Apr. 12th)

Here are Quiz 12 problems and solutions.
§4.4 p.305- # 7, 8, 9, 11, 12, 14, 17, 18, 21, 24, 25, 27
§4.5 p.314- # 10, 11, 13, 16, 17, 18 (newly added), 21, 22, 23, 24, 25, 26

### Quiz 13 (Apr. 19th) The last quiz

Here is the first problem of Quiz 13 and its solution.
Here is the second problem of Quiz 13 and its solution.
§4.6 p.324- # 8, 10, 16, 20, 21, 29, 36-42
§4.7 p.336- # 1-12, 17, 26, 43

## Question?

If you have any question about linear algebra, ask the question in the Q&A forum.