Math 2568 Linear Algebra Spring 2018

Some of the contents and links were deleted after the final exam.

Informaiton

Instructor: Yu Tsumura

When:

• MWF 12:40PM - 1:35PM MATH 2568-0070 (32366)
• or MWF 1:50-2:45 MATH 2568-0085 (32370)

(Note:4568 is 2568 for graduate students.)

Where: Stillman Hall 235

News
4/22 The office hour on Monday (4/23) is the last office hour of the semester.
4/22 Some of the typos in Workbook were corrected.
4/1 Office hour on 4/3 is 11:15-11:45
2/26 Errata of workbook was updated.
2/19 Office hour on 2/20 is 11:00-11:30.
2/10 No Office hour on 2/12 for a make-up exam.
1/28 Office hours on 1/29 and 2/4 starts at 10:15. Office hour on 1/30 starts at 11:00.
1/25 The second midterm date has been changed to March 28th Wednesday. Homework 10 is due March 30th.
1/22 The syllabus has been updated (Office hours and grader's info)

Textbook

You do not have to buy any textbook for my sections.

Resources

• I will give Linear Algebra Workbook handouts in class.
Typos in Workbook.

Let me know if you find a typo or error.

• p.4 $E_1 \leftrightarrow E_2$ should be $E_1 \leftrightarrow E_3$.
• p.6 $E_{1}\leftrightarrow E_{1}$ should be $E_{1}\leftrightarrow E_{3}$.
• p. 17 Theorem 5.1.1 (3) should be “where $O$ is the m x n zero matrix”
• p.19 Theorem 5.1.4 (1) should be $(A+B)^{T}=A^{T}+B^{T}$
• p.27 Theorem 7.0.1 should be “If $p>n$”.
• p. 28 the second matrix in Example 7.2 should be labeled B
• p.44 $\R^n$ should be $\R^m$.
• p.47 Definition 12.1.2. The row vectors are $\mathbf{a}_{1},\mathbf{a}_{2},\dots,\mathbf{a}_{m}$, not $\mathbf{a}_{1},\mathbf{a}_{2},\dots,\mathbf{a}_{n}$.
• p.67 Exercise 108 (h), $C[-1, -1]$ should be $C[-2, 2]$.
• p.69 Example 21.2 should be $E_{11}= \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} ,\; E_{12}= \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} ,\; E_{21}= \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} ,\; E_{22}= \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} .$
• p.73 Example 23.1 should say “Let $V$ be the set…”
• p.89 should be $F(\mathbf{e}_{3})=F\left( \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \right)= \begin{bmatrix} 0 \\ -1 \end{bmatrix}$
• p. 105 Exercise 174 should say “characteristic polynomial given by”
• p. 116 top of page says $SKS^{-1}$ multiple times when it should say $SDS^{-1}$

• Linear Algebra Problems and Solutions.
• This is Linear Algebra by Chrichton Ogle

Final Exam

12:40 section: May.1st(T) 12:00-1:45pm in the typical class room.

1:50 section: Apr.25th(W) 6:00-7:45pm in Hitchcock Hall 031

Homework

Homework 1 Due 1/17 in class

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

• Lecture 1. 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. Exercise 5, 6, 7
• Lecture 3. Exercise 12, 13

Solution 1

• Solution 5
• Solution 6
• Solution 7
• Solution 12
• Solution 13
• Comment from the grader: For Lecture 3 Exercise 12, one needs to reduce the matrix to row echelon form. In doing so, one should avoid dividing a row by zero. For example, multiplying a row by 1/a is not proper when a=0.
For Lecture 3 Exercise 13(i), do not forget to consider the case when the system is inconsistent.

Homework 2 Due 1/24 in class

Do and submit your solutions to the following problems.

• Lecture 4. Exercise 16, 17, 18, 22
• Lecture 5. Exercise 23, 24, 26, 28
• Note that a diagonal matrix is a matrix whose non-diagonal entries are all $0$. For example,
$\begin{bmatrix} 2&0\\ 0& 3\end{bmatrix}, \begin{bmatrix} 1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & -3\end{bmatrix}$
are diagonal matrices.

Solution to Homework 3

Comment from the grader: For lecture 5 exercise 23, notice that $A^{T}-(A-B)^{T}=A^{T}-(A^{T}-B^{T})=B^{T}$, one could save some computation.
For lecture 5 exercise 28, notice that $Au=-u$, so $A^{n}u=(-1)^{n}u$.

Homework 3 Due 1/31 in class

Do and submit your solutions to the following problems.

• Lecture 6. Exercise 31, 35, 37
• Lecture 7. Exercise 39, 40, 41, 42, 45
• Lecture 8. Exercise 50, 51, 52

Homework 4 Due 2/7 in class

Do and submit your solutions to the following problems.

• Lecture 9. Exercise 55, 56, 58
• Lecture 10. Exercise 64, 65, 66
• Lecture 11. Exercise 71, 73, 74

Solution to Homework 4

Comment from the grader: For exercise 58(b), suppose $A$ is nonsingular, then $A^{-1}$ exists. From $A^{2}=AB+2A$, we get $A=A^{-1}A^{2}=A^{-1}(AB+2A)=B+2I$. Some Students got the wrong answer $B+2$ which doesn’t make sense. $A^{-1}A$ is equal to the 3 by 3 identity matrix $I$, not the real number 1.
For exercise 71(f), the set $S=\{ a, b, c \}$ contains 3 vectors, but that doesn’t mean span of $S$ is a subspace of $\mathbb{R}^{3}$. In fact, for all parts of exercise 71, since $S \subset \mathbb{R}^{2}$, the span of $S$ is a subspace of $\mathbb{R}^{2}$.

Homework 5 Due 2/14 in class

Do and submit your solutions to the following problems.

• Lecture 12. Exercise 76, 77, 78, 79, 80, 81

Solution to Homework 5

Comment from the grader: For exercise 79, we form the augmented matrix and reduce it to row echelon form. Then the consistency condition gives the defining equation of $R(A)$. Some students think that the last column of the reduced augmented matrix gives the general form of a vector in $R(A)$, that’s not right.

Homework 6 Due 2/21 in class

Do and submit your solutions to the following problems.

• Lecture 15. Exercise 85, 86, 87
• Lecture 16. Exercise 90, 91, 92
• Lecture 17. Exercise 96, 97

Solution to Homework 6

Comment from the grader: For exercise 87, be aware of the difference between spanning set and basis. A basis is a linearly independent spanning set, while in general a spanning set need not be linearly independent. So it is possible that a subset of a spanning set spans the same space.

Homework 7 Due 2/28 in class

Do and submit your solutions to the following problems.

• Lecture 18. Exercise 101, 102, 103 (You may or may not assume the vector $\mathbf{v}$ is nonzero.)
• Lecture 19. Exercise 106, 107 (Only 2, 3, 5)
• Lecture 20. Exercise 108 (d-i), 111 (b), 113

Solution to Homework 7

Comment from the grader: A lot of students have trouble doing exercise 107, please see the proved parts in lecture notes to learn how to use axioms, try again before referring to prof. Tsumura’s online solution. Also, I want to point out one common mistake. Part 5 of exercise 107 asks one to deduce $a=0$ or $v=0$ from $av=0$, a lot of students wrongly conclude this from part 3 and 4, which proves the opposite direction.

Homework 8 Due 3/7 in class

Do and submit your solutions to the following problems.

• Lecture 21. Exercise 115
• Lecture 22. Exercise 116, 117, 118
• Lecture 23. Exercise 121, 122, 125

Homework 9 Due 3/21 in class

Do and submit your solutions to the following problems.

• Lecture 24. Exercise 127, 129, 130
• Lecture 25. Exercise 133, 134 (The norm of a vector is the same as the length of the vector), 137
• Lecture 26. Exercise 139, 140, 141

Homework 10 Due 3/30 in class

Do and submit your solutions to the following problems.

• Lecture 27. Exercise 148, 149, 151 (Do these before midterm 2)
• Lecture 28. Exercise 153, 155, 156
• Lecture 29. Exercise 159, 160, 161

Homework 11 Due 4/4 in class

Do and submit your solutions to the following problems.

• Lecture 32. Exercise 167, 168, 169 (Part (c) optional), 170 (Part 1 and 3)

Homework 12 Due 4/11 in class

Do and submit your solutions to the following problems.

• Lecture 33 Redo midterm 2 (No need to submit this)
• Lecture 34 Exercise 172, 174, 175
• Lecture 35 Exercise 176, 177, 179

Homework 13 Due 4/18 in class

Do and submit your solutions to the following problems.

• Lecture 36 Exercise 180, 181, 182
• Lecture 37 Exercise 186, 188 and Practice 36.2.3
• Lecture 38 Exercise 195, 196, 197