# 2568 Linear Algebra Autumn 2017

The course website for Spring 2018 is here

Instructor: Yu Tsumura

Where: Macquigg Laboratory 160

When: MWF 1:50-2:45

News
12/23 Some links were disable after the semester.
11/8 Midterm Problems and Solutions were posted.
11/6 Lecture Notes for Gabriel Valenzuela Vasquez’s class(9:10) and the solution of the problem you asked.
11/2 No Office hour on Monday Nov.6th
10/18 2017 Spring Midterm 2 problems and solutions were added.
9/25 Links to Midterm 1 problems and solutions were added.
9/25 Wednesday office hours moved to Fridays 12:10-1:40.
9/1 The information about the grader is added in the syllabus.
8/10 The course webpage for 2568 (Yu Tsumura) was created.

## 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

## Lecture Notes

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

## Midterm 2 Information

We will have midterm 2 on 11/3 Friday in class.

The exam will cover the materials we studied in Chapter 3 and 5 of the textbook.

### More practice problems for midterm 2

Check out the list of linear algebra problems and study problems from Chapter 3 and 5.

The followings are past exam problems from Spring 2017.

## 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

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

### 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

#### 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

Comment from 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.
• #### 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

Comment from the grader:
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,
• #### 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

Comment from 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

#### Solution 6

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

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
• #### 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

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)
• #### 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

#### Solution 8

Comment from the grader: For Ex 23 in section 5.2, when you formulate the proof, you should explain in each step, which property in Definition 1 is used.

For Ex 4 in Section 5.3, many of you know that $W$ is not a subspace of $V$, in particular, $W$ doesn't satisfy (S2). However, to prove $W$ fails in (S2), you should prove there is some $A$ and $B$ in $W$, but $A+B$ is not in $W$, that is, you should provide a counterexample. Please see the solution for details.

### 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 $A_3=\begin{bmatrix} 0 & 0\\ 2& 0 \end{bmatrix}$)
• Lecture 24 §5.4 p.387- #23, 30 (same remark for $A_3$ as in #29), 34 (a hint for #34)
• #### Extra (Not submit these)

• Lecture 22, 23, 24 §5.4 p.387- #1, 5, 9, 10, 11, 12, 14, 15, 18, 27, 36, 37

#### Solution 9

Comment from the grader: Most of you know that Ex 30 turns into solving a system of nonhomogeneous linear equations, and we learned that in Chapter 1, but many of you did something wrong in the computation. Please try again before referring to the solution.

### HW 10 (Due 11/1 in class)

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

• Lecture 25 §3.6 p.224- #8, 16, 20
• Lecture 26 §3.7 p.239- #10, 14, 19
• Lecture 27 §3.7 p.239- #20 (c), 22, 29
• #### Extra (Not submit these)

• Lecture 25 §3.6 p.224- #1, 5, 12, 13, 21, 28
• Lecture 26, 27 §3.7 p.239- #1, 2, 3, 4, 6, 13, 18, 20, 21, 25, 30

#### Solution 10

Comment from the grader: "For Ex 22 in section 3.7, to describe the linear transform T, we should describe $T(e_1)$ and $T(e_2)$, where $\{e_1, e_2\}$ is the standard basis for $\mathbb{R}^2$. Now the problem gives us how $T$ does on $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ and $\begin{bmatrix} 1 \\ -1 \end{bmatrix}$, so we need to express the standard basis vectors as linear combinations of vectors in the new basis, and then use the linearity property of $T$ (i.e., the equation (6)). Please see the solution for details. "

### HW 11 (Due 11/8 in class)

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

• Lecture 28 §4.1 p.279- #6, 10, 17
• #### Extra (Not submit these)

• Lecture 28 §4.1 p.279- #1, 3, 7, 13, 18, 19

#### Solution 11

Comment from the grader: For Ex 17, the problem wants us to show for symmetric $2 \times 2$ matrix, there is an eigenvalue (Note that this is not necessarily true for a non-symmetric matrix, for example the matrix $\begin{bmatrix} 0 & 1\\ -1& 0 \end{bmatrix}$ does not have a real eigenvalue). Showing there is an eigenvalue is the same as showing there is a root for the quadric $\lambda^2-(a+d)\lambda+(ad-b^2)$. But the discriminant of the quadric is $(a-d)^2+4b^2$, which is greater or equal to zero, meaning the quadric has at least one root. This is what we want to show. Please refer to solution for a complete proof.

### HW 12 (Due 11/15 in class)

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

• Lecture 31 §4.2 p.288- #14, 18, 21, 29
• Lecture 32
Problem A. Let $T:\R^2\to \R^2$ be a linear transformation such that it maps the vectors $\mathbf{v}_1, \mathbf{v}_2$ as indicated in the figure below.

Find the matrix representation $A$ of the linear transformation $T$.

Problem B. An $n\times n$ matrix $A$ is called orthogonal if $A^{\trans}A=I$.
Let $V$ be the vector space of all real $2\times 2$ matrices.
Consider the subset
$W:=\{A\in V \mid \text{A is an orthogonal matrix}\}.$
Prove or disprove that $W$ is a subspace of $V$.
Problem C. Let $C[-2\pi, 2\pi]$ be the vector space of all real-valued continuous functions defined on the interval $[-2\pi, 2\pi]$.
Consider the subspace $W=\Span\{\sin^2(x), \cos^2(x)\}$ spanned by functions $\sin^2(x)$ and $\cos^2(x)$.
(a) Prove that the set $B=\{\sin^2(x), \cos^2(x)\}$ is a basis for $W$.

(b) Prove that the set $\{\sin^2(x)-\cos^2(x), 1\}$ is a basis for $W$.
(Hint: Use the coordinate vectors with respect to the basis $B$ from part (a).)

#### Extra (Not submit these)

• Lecture 31 §4.2 p.288- #17, 19, 22, 24, 26, 30
• Lecture 32 Review and redo midterm 2 problems. Compare your solutions with mine.

#### Solution 12

Solutions to Problem A, Problem B, Problem C

Comment from the grader: To show $\{v_1,...,v_k\}$ form a basis for a vector space $W$, one need to verify three things: 1. $\{v_1,...,v_k\}\in W$ ; 2. $\{v_1,...,v_k\}$ are linearly independent; 3. $\dim(W)=k$

### HW 13 (Due 11/20 Monday in class)

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

• Lecture 33 §4.4 p.305- #3, 8, 14
• Lecture 34 §4.4 p.305- #9, 18, 19
• Lecture 35 §4.5 p.314- #12, 16, 18

#### Extra (Not submit these)

• Lecture 33 §4.4 p.305- #1, 9, 11, 15, 16
• Lecture 34 §4.4 p.305- #12, 17, 21, 25
• Lecture 35 §4.5 p.314- #3, 9, 11, 17, 19, 21, 22

#### Solution 13

Comment from the grader: For Ex 18 in section 4.5, we want to find $A^{10}X$, where $X=\begin{bmatrix} 0 \\ 9 \end{bmatrix}$, which is hard to compute directly, but we know the same expression is easy to compute when $X$ is an eigenvector of $A$. After finding the eigenvectors $\begin{bmatrix} 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 2 \\ 5 \end{bmatrix}$ of $A$, we simply need to express $\begin{bmatrix} 0 \\ 9 \end{bmatrix}=a\begin{bmatrix} 1 \\ 1 \end{bmatrix}+b\begin{bmatrix} 2 \\ 5 \end{bmatrix}$ and find coefficients $a, b$, and everything will follow.

### HW 14 (Due 11/29 Wednesday in class)

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

• Lecture 36 §4.5 p.314- #17, 22 and §4.6 p.324- #21, 23, 38

#### Extra (Not submit these)

• Lecture 36 §4.6 p.324- #16, 19, 22, 29, 39, 40, 41, 42
• Please take the textbook survery.

### HW (No submit)

The following problems will not be collected and graded. However, you need to do these problems to prepare for the final exam.

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