### Course Information

Some links were disabled after the semester.

Instructor: Yu Tsumura

Where: Stillman Hall 235

When: MWF 12:40-1:35

### 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.

1/30 Syllabus is updated. (The grader's info was added.)

1/9 First Class

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

## 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.

### More practice problems for Final Exam

- Check out the list of linear algebra problems
- Here are past exam problems and solution at the Ohio State University
- Here are linear algebra exam problems from various universities.
- How to diagonalize a matrix. Step by step explanation.

### List of Quiz Problems of Linear Algebra

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

- Quiz 1. Gauss-Jordan elimination / homogeneous system.
- Quiz 2. The vector form for the general solution / Transpose matrices.
- Quiz 3. Condition that vectors are linearly dependent/ orthogonal vectors are linearly independent
- Quiz 4. Inverse matrix/ Nonsingular matrix satisfying a relation
- Quiz 5. Example and non-example of subspaces in 3-dimensional space
- Quiz 6. Determine vectors in null space, range / Find a basis of null space
- Quiz 7. Find a basis of the range, rank, and nullity of a matrix
- Quiz 8. Determine subsets are subspaces: functions taking integer values / set of skew-symmetric matrices
- Quiz 9. Find a basis of the subspace spanned by four matrices
- Quiz 10. Find orthogonal basis / Find value of linear transformation
- Quiz 11. Find eigenvalues and eigenvectors/ Properties of determinants
- Quiz 12. Find eigenvalues and their algebraic and geometric multiplicities
- Quiz 13 (Part 1). Diagonalize a matrix.
- Quiz 13 (Part 2). Find eigenvalues and eigenvectors of a special matrix

## 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 $\R^n$."

### More practice problems for midterm 2

- Check out the list of linear algebra problems and study problems from Chapter 3 and 5.
- Here are past exam problems and solution at the Ohio State University. But remark that some of them are not problems of Chapter 3 and 5. The problems contains exam problems of midterm exams and final exam at OSU.
- Here are linear algebra exam problems from various universities. The same remark as above applies here as well.
- 10 examples of subsets that are not subspaces of vector spaces

### Midterm Exam 2 Problems and Solutions

- True of False Problems and Solutions: True or False problems of vector spaces and linear transformations
- Problem 1 and its solution: See (7) in the post "10 examples of subsets that are not subspaces of vector spaces"
- Problem 2 and its solution: Determine whether trigonometry functions $\sin^2(x), \cos^2(x), 1$ are linearly independent or dependent
- Problem 3 and its solution: Orthonormal basis of null space and row space
- Problem 4 and its solution: Basis of span in vector space of polynomials of degree 2 or less
- Problem 5 and its solution: Determine value of linear transformation from $\R^3$ to $\R^2$
- Problem 6 and its solution: Rank and nullity of linear transformation from $\R^3$ to $\R^2$
- Problem 7 and its solution: Find matrix representation of linear transformation from $\R^2$ to $\R^2$
- Problem 8 and its solution: Hyperplane through origin is subspace of 4-dimensional vector space

## 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.

### More practice problems for midterm 1

- Check out the list of linear algebra problems and study problems from Chapter 1.
- Here are past exam problems and solution at the Ohio State University. But remark that some of them are not problems of Chapter 1. The problems contain exam problems of midterm exams and final exam at OSU.
- Here are linear algebra exam problems from various universities. The same remark as above applies here as well.
- Summary: possibilities for the solution set of a system of linear equations

### Midterm 1 problems and solutions

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

The problems and solutions.

- Problem 1 and its solution: Possibilities for the solution set of a system of linear equations
- Problem 2 and its solution: The vector form of the general solution of a system
- Problem 3 and its solution: Matrix operations (transpose and inverse matrices)
- Problem 4 and its solution: Linear combination
- Problem 5 and its solution: Inverse matrix
- Problem 6 and its solution: Nonsingular matrix satisfying a relation
- Problem 7 and its solution: Solve a system by the inverse matrix
- 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 Introduction to Matrices and Systems of Linear Equations
- §1.2 Echelon Form and Gauss-Jordan Elimination
- §1.3 Consistent Systems of Linear Equations
- §1.5 Matrix Operations
- §1.6 Algebraic Properties of Matrix Operations
- §1.7 Linear Independence and Nonsingular Matrices
- §1.7 Linear Independence and Nonsingular Matrices Part 2
- §1.7 Linear Independence and Nonsingular Matrices Part 3
- §1.9 Matrix Inverse and Their Properties
- §1.9 Matrix Inverse and Their Properties Part 2
- §3.2 Vector Space Properties of $\R^n$
- §3.3 Examples of Subspaces
- §3.3 Examples of Subspaces Part 2
- Midterm 1
- §3.3 Examples of Subspaces Part 3
- Review of Midterm 1
- §3.4 Bases for Subspaces
- §3.4 Bases for Subspaces Part 2
- §3.5 Dimensions
- §3.5 Dimensions Part 2
- §5.2 Vector Spaces
- §5.3 Subspaces
- §5.3 Subspaces Part 2
- §5.4 Linear Independence, Bases, and Coordinates
- < §5.4 Linear Independence, Bases, and Coordinates Part 2
- §5.4 part 3 & §3.6 Orthogonal Bases for Subspaces
- §3.6 Orthogonal Bases for Subspaces Part 2 & §3.7 Linear Transformation from $\R^n$ to $\R^m$
- §3.7 Linear Transformation from $\R^n$ to $\R^m$. Part 2
- §4.1 The Eigenvalue Problem for $(2\times 2)$ Matrices
- §4.2 Determinants and the Eigenvalue Problem
- §4.2 Determinants and the Eigenvalue Problem Pat 2
- Exam 2.
- §4.4 Eigenvalues and the Characteristic Polynomial
- §4.5 Eigenvectors and Eigenspaces
- §4.5 Eigenvectors and Eigenspaces Part 2
- §4.6 Complex Eigenvalues and Eigenvectors
- §4.7 Similarity Transformation and Diagonalization
- §4.7 Similarity Transformation and Diagonalization part 2
- §4.7 Similarity Transformation and Diagonalization part 3
- Practice problems for the final exam 1
- Practice problems for the final exam 2
- 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).

Grading scheme

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.

If you want to help other students, please answer questions.