# 2568 Linear Algebra Autumn 2017

Instructor: Yu Tsumura

Where: Macquigg Laboratory 160

When: MWF 1:50-2:45

News
9/25 Wednesday office hours moved to Fridays 12:10-1:40.
8/10 The course webpage for 2568 (Yu Tsumura) was created.

## Textbook

The required textbook is

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

## Lecture Notes

1. Lecture Notes 1 (Preview) §1.1 Introduction to Matrices and Systems of Linear Equations
2. Lecture Notes 2 (Preview) §1.2 Echelon Form and Gauss-Jordan Elimination
3. Lecture Notes 3 (Preview) §1.3 Consistent Systems of Linear Equations
4. Lecture Notes 4 (Preview) §1.5 Matrix Operations
5. Lecture Notes 5 (Preview) §1.6 Algebraic Properties of Matrix Operations
6. Lecture Notes 6 (Preview) §1.7 Linear Independence and Nonsingular Matrices
7. Lecture Notes 7 (Preview) §1.7 Linear Independence and Nonsingular Matrices Part 2
8. Lecture Notes 8(Preview) §1.9 Matrix Inverse and Their Properties
9. Lecture Notes 9 (Preview) §1.9 Matrix Inverse and Their Properties Part 2
10. Lecture Notes 10 (Preview) §3.2 Vector Space Properties of $\R^n$
11. Lecture Notes 11 (Preview) §3.3 Examples of Subspaces
12. Lecture Notes 12 (Preview) §3.3 Examples of Subspaces Part 2
13. Midterm 1. Bring the Buck ID with you.
14. Lecture 14. Review of Midterm 1.
15. Lecture Notes 15 (Preview) §3.4 Bases for Subspaces
16. Lecture Notes 16 (Preview) §3.4 Bases for Subspaces Part 2
17. Lecture Notes 17 (Preview) §3.4 Part 3 and §3.5 Dimensions
18. Lecture Notes 18 (Preview)§3.5 Dimensions Part 2
19. Lecture Notes 19 (Preview)§5.2 Vector Spaces
20. Lecture Notes 20 (Preview) §5.2 Part 2 & §5.3 Subspaces
21. Lecture notes 21 (Preview) §5.3 Subspaces Part 2
22. Lecture notes 22 (Preview) §5.4 Linear Independence, Bases, and Coordinates
23. Lecture notes 23 (Preview) §5.4 Linear Independence, Bases, and Coordinates Part 2
24. Lecture notes 24 (Preview) §5.4 part 3 & §3.6 Orthogonal Bases for Subspaces
25. §3.6 Orthogonal Bases for Subspaces Part 2 & §3.7 Linear Transformation from $\R^n$ to $\R^m$
26. §3.7 Linear Transformation from $\R^n$ to $\R^m$. Part 2
27. §4.1 The Eigenvalue Problem for $(2\times 2)$ Matrices
28. §4.2 Determinants and the Eigenvalue Problem
29. §4.2 Determinants and the Eigenvalue Problem Pat 2
30. Exam 2.

## Midterm 1 Information

We will have midterm 1 on 9/22 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 (including extra 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.

The followings are past exam problems from Spring 2017.

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

## Homework Assignments

Please use the cover sheet for the homework assignments.

(You may reuse the cover sheet. Please remove the old staple neatly and staple again.)

### HW 1 (Due 8/30 in class)

Do and submit your solutions of the following problems from the textbook.

• Lecture 1 §1.1 p.12- #2, 20, 28.
Also take a quiz here. (No need to submit this quiz. This is just for your practice. You can take the quiz as many times as you want.)
• Lecture 2 §1.2 p.26- #18, 28, 44.
Note that for #44, "the matrix $A$ is row equivalent to the matrix $I$" means that if you apply several elementary row operations, then $A$ is transformed into $I$.

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

#### Extra (Not submit these)

• Lecture 1 §1.1 #3, 4, 11, 24, 32
• Lecture 2 §1.2 #3, 9, 15, 17, 23, 37, 54

#### Solution 1

The solutions of the homework #1 written by the grader.

Comment from the grader: I have a comment for Ex 44:To show matrix $A$ is row equivalent to identity matrix $I$, please only rely on the assumption $b-cd$ is nonzero (so you can divided by this number) in your proof, so you mayor divide rows by $d$ or $b$, for example. Besides, you cannot multiply $R_1$ by $c$, because if $c$ is zero, then this is not proper row reduction.

### HW 2 (Due 9/6 in class)

Do and submit your solutions of the following problems from the textbook.

Please use the cover sheet (can be found above).
(You may reuse the cover sheet from HW 1.)

#### Extra (Not submit these)

• Lecture 3 §1.3 p.38- #1, 8, 10, 12, 16, 20, 24.
• Lecture 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
• Lecture 5 §1.6. p.69- #13, 15, 27, 28, 31, 34, 42, 46, 49

#### Solution 2

The solutions of the homework #2 written by the grader.

### HW 3 (Due 9/13 in class)

Do and submit your solutions of the following problems from the textbook.

• Lecture 6 §1.7 p.78. #12, 38, 51 and the following problem.
Let
$\mathbf{v}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} 1 \\ a \\ 5 \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 0 \\ 4 \\ b \end{bmatrix}$ be vectors in $\R^3$.
Determine a condition on the scalars $a, b$ so that the set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is linearly dependent.
• Lecture 7 §1.7 p.78.#23, 44, 46, 48, 56

Please use the cover sheet (can be found above).
(You may reuse the cover sheet from HW 1.)

#### Extra (Not submit these)

• Lecture 6 §1.7 p.78. 4, 11, 13, 30, 31, 39, 46, 49, 50, 52
• Lecture 7 §1.7 p.78. 40, 54, 55, 57

#### Solution 3

The solution of the above problem is given ↴
Determine a Condition on $a, b$ so that Vectors are Linearly Dependent

Solutions #3 written by the grader

For Ex 46 (a), A geometric picture of this problem is that in a plane, any three vectors must be linearly dependent. The explanation is the following: the linear dependence relation of the three vectors is equivalent to solving a homogeneous linear system with 3 variables, and 2 equations, which means, the system always admits a nonzero solution, and this proves that three vectors are linearly dependent.

### HW 4 (Due 9/20 in class)

Do and submit your solutions of the following problems from the textbook.

• Lecture 8 §1.9 p.102- #20, 28, 48
• Lecture 9 §49, 50, 67
• Lecture 10 Do and submit 8 problems of Midterm 1 from Spring 2017 listed above. The first problem is here.
• Please use the cover sheet (can be found above).
(You may reuse the cover sheet from HW 1.)

#### Extra (Not submit these)

• Lecture 8 §1.9 p.102- #1, 9, 13, 23, 27, 29, 30
• Lecture 9 §1.9 p.102- #39, 40, 51, 52, 54, 55, 56, 72, 74

#### Solution 4

Solutions #4 written by the grader
Note: For #50, once you obtain the matrix $A$, you need to check whether $A$ is actually a nonsinguylar matrix.

Graded problems: Ex28 (3 points), Ex 50(4 points), and Ex 67(3 points).

For Ex 28, to find $\lambda$ such that the matrix is invertible is the same as the matrix is nonsingular. As what as we did before, we do row reduction to make the matrix into echelon form (not need to be reduced echelon form) and such that the first and second rows contain only numbers, while the 3rd row is $(0,0,\lambda-34/7)$. Now, the matrix is invertible is the same as saying $\lambda-34/7$ not equal to $0$. Details see the answer.

For Ex 50, from the equation $A^2=AB+2A$, multiplying the $A^{-1}$ on the left, we get $A^{-1}A^2= A^{-1}AB+ 2A^{-1}A$, this is $A=B+2I$, but if you multiply $A^{-1}$ on the right, you will get: $A=ABA^{-1}+2I$. Recall by an exercise in HW2, we know in general $AB$ is not equal to $BA$, and so in general $ABA^{-1}$ is not equal to $B$, and this equation is not what we want.

### HW 5 (Due 9/27 in class)

Do and submit your solutions of the following problems from the textbook.

• Lecture 11 §3.2 p.174- #8, 10, 18,
• Lecture 12 §3.3 p.186- #30, 36, 38,
• Please use the cover sheet (can be found above).
(You may reuse the cover sheet from HW 1.)

#### Extra (Not submit these)

• Lecture 11 §3.2 p.174- # 2, 3, 11, 12, 13, 16, 19, 30, 31
• Lecture 12 §3.3 p.186- #4, 6, 9, 15, 17, 20, 22, 25, 26, 34, 40, 43, 46, 49

#### Solution 5

Solutions #5 written by the grader

Graded problems: problem 8 of section 3.2 and problem 30, 38 in section 3.3.

For Ex 8, verify that both $(0,1)$ and $(1,0)$ are in the set $W$, but their addition, $(1,1)$ is not in $W$.

For Ex 30, when you want to describe the range of a matrix $\calR(A)$ algebraically, you should try to write it as linear combinations of some vectors. For example, in this problem, it is a linear combination of $[0,1]$ and $[1,0]$, or it is the whole space $\R^2$.

### HW 6 (Due 10/4 in class)

Do and submit your solutions of the following problems from the textbook.

• Lecture 14
Problem A. Prove that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is also nonsingular.
(You may only use the definition of a nonsingular matrix.)
Problem B. Let $B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 2 & 1 & -4 \end{bmatrix}.$ Find a nonsingular matrix $A$ satisfying
$A^2=AB+2A$ if exists. If you think such a nonsingular matrix does not exist, exlain why.
Problem C. Let $A$ be a $6\times 6$ matrix and suppose that $A$ can be written as
$A=BC,$ where $B$ is a $6\times 5$ matrix and $C$ is a $5\times 6$ matrix.

Prove that the matrix $A$ cannot be invertible.

• Lecture 15 §3.4 p.200- #25, 26, 32
• Lecture 16 §3.4 p.200- #12, 18, 24
• Please use the cover sheet (can be found above).
(You may reuse the cover sheet from HW 1.)

#### Solution 6

The solutions of Problem A, Problem B, and Problem C.

Solutions #6 written by the grader

Comment from the grader: For Ex B online, you should be careful to check if the matrix $A$ you get is indeed nonsingular, otherwise you may not multiply by $A^{-1}$ on both side of equation from the start. You should compare a similar homework problem in HW 4. Also, please refer to Dr. Tsumura's solution online for this problem.

For Ex 24, many of you are wrong in part (a). To be more precise, many of you find the basis for $\Span(S)$ containing 2 vectors, and in part (b) find another basis containing 3 vectors. This is a contradiction, since number of vectors in a basis for a space should be the same. Please see solution to see the details.

### HW 7 (Due 10/11 in class)

Do and submit your solutions of the following problems from the textbook.

• Lecture 17 §3.5 p. 212- #24, 27 (b), 29
• Lecture 18 §3.5 p. 212- #18, 31, 34,
• Lecture 19 No assinments for this lecture
• Please use the cover sheet (can be found above).
(You may reuse the cover sheet from HW 1.)

#### Extra (Not submit these)

• Lecture 17 §3.5 p. 212- #4, 6, 8, 12, 18, 28,
• Lecture 18 §3.5 p. 212- #35, 37, 38, 39

#### Solution 7

Solutions #7 written by the grader

Comment from the grader: In Ex 34, showing a $3\times 4$ matrix has linearly dependent columns is the same as showing a system of 3 linear equations in 4 variables admits infinitely many solutions, which is what we learned in Chapter 1, so is done. Another way to see is that any 4 vectors in $\R^3$ are always linearly dependent (Theorem 9, part I).
Note that if you get points off, you may fail to explain why the system admits a nonzero solution, so you may not start from the equation $c_{1}A_{1}+c_{2}A_{2}+c_{3}A_{3}+A_{4}=\mathbf{0}$.

### HW 8 (Due 10/18 in class)

Do and submit your solutions of the following problems from the textbook.

• Lecture 20 §5.2 p.366- #8, 22, 23,
• Lecture 21 §5.3 p.373 #4, 8, 10, 22, 26, 31 (d)
• Please use the cover sheet (can be found above).
(You may reuse the cover sheet from HW 1.)

#### Extra (Not submit these)

• Lecture 20 §5.2 p.366- #10, 11, 13, 16, 17, 21, 26
• Lecture 21 §5.3 p.373 3, 7, 12, 13, 17, 19, 23, 27

### HW 9 (Due 10/25 in class)

Do and submit your solutions of the following problems from the textbook.

• Lecture 22 §5.4 p.387- #32, 33
• Lecture 23 §5.4 p.387- #22, 24, 26, 29 (Instead of $A_3$ in the textbook, use $A_3=\begin{bmatrix} 0 & 0\\ 2& 0 \end{bmatrix}$)
• Please use the cover sheet (can be found above).
(You may reuse the cover sheet from HW 1.)

#### Extra (Not submit these)

• Lecture 22, 23 §5.4 p.387- #1, 5, 9, 10, 11, 12, 14, 15, 18, 23, 27, 30, 34, 36, 37

## Linear Algebra Q&A

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