Linear Algebra and Probability, Spring 2018 
1  Mon, 22 Jan  Download the Course Syllabus.
Homework: Read pp. 3—11, and do Exercises #1—5, 7. Visit last year's website for sample quizzes, exams, and much, much more! 
2  Wed, 24 Jan 
Homework: Finish reading Section 2.0 (but skip all the trigonometric derivations on pp. 14—16). Do problems 9—10, 12—16, and 18. Homework answers: 1. (a) $x=2.$ (b) $x=0, 3.$ (c) $x=3,3.$ (d) all real numnbers. 2. False (consider a $0$ scalar multiple). 3. True (no difficulties with nonzero scalar multiples). 4. $x=1+2t, y=4+2t$ (other answers are possible). 5. $x=32t, y=1+t$ (other answers are possible). 
3  Fri, 26 Jan 
Read Section 2.1. Do p. 25, Exercises #1(a)—(c), 2, 3(a)—(c), 4, 5. There will be a Homework Quiz next Wednesday! Answers to selected homework Exercises (starting on p. 13): 10. $\left(\begin{matrix}2\\5\end{matrix}\right).$ 12. $\left(\begin{matrix}1\\2\end{matrix}\right).$ 13. $45^\circ.$ 14. $135^\circ.$ 15. $\left(\begin{matrix}3/10\\1/10\end{matrix}\right).$ 16. $\left(\begin{matrix}1\\2\end{matrix}\right).$ 18. $\sqrt5.$ 
4  Mon, 29 Jan 
Nick's office hours tonight are from 5:00—7:00 in Room 315 in the library. Answers to yesterday's homework: 1(a). $x'=x,\quad y'=0.$ 1(b). $x'=0,\quad y'=y.$ 1(c). $x'=\dfrac{xy}2,\quad y'=\dfrac{yx}2.$ 2. $a'=\dfrac{4a+10b}{29},\quad b'=\dfrac{10a+25b}{29}.$ 3(a). $x'=x,\quad y'=y.$ 3(b). $x'=x,\quad y'=y.$ 3(c). $x'=y,\quad y'=x.$ 4. $a'=\dfrac{21a+20b}{29},\quad b'=\dfrac{20a+21b}{29}.$ 5a. $\left[\begin{matrix}0&1\\1&0\end{matrix}\right].$ 5b. $\left[\begin{matrix}1&0\\0&1\end{matrix}\right].$ 5c. $\left[\begin{matrix}1&0\\0&1\end{matrix}\right].$ Download a useful summary of linear and affine transformations. Today's homework:

5  Wed, 31 Jan 
Answers to yesterday's HW (questions involving formulas): 1. $f\left(\begin{matrix}x\\ y\end{matrix}\right)=\left[\begin{matrix}2&0\\0&1\end{matrix}\right]\left(\begin{matrix}x\\ y\end{matrix}\right)+\left(\begin{matrix}2\\ 1\end{matrix}\right).$ 2. $f\left(\begin{matrix}x\\ y\end{matrix}\right)=\left[\begin{matrix}1&0\\0&1\end{matrix}\right]\left(\begin{matrix}x\\ y\end{matrix}\right)+\left(\begin{matrix}1\\ 0\end{matrix}\right).$ 6. $f\circ g\left(\begin{matrix}x\\ y\end{matrix}\right)=\left(\begin{matrix}y\\ x\end{matrix}\right)$ and $g\circ f\left(\begin{matrix}x\\ y\end{matrix}\right)=\left(\begin{matrix}y\\ x\end{matrix}\right).$ Here is the fractal we looked at in class today. Today's homework:

6  Fri, 2 Feb 
Answers to yesterday's homework: 1. $f\left(\begin{matrix}x\\ y\end{matrix}\right)=\left[\begin{matrix}1&0\\0&1\end{matrix}\right]\left(\begin{matrix}x\\ y\end{matrix}\right)+\left(\begin{matrix}3\\ 2\end{matrix}\right).$ 2. $\left[\begin{matrix}0&3\\2&0\end{matrix}\right].$ 3. $f\left(\begin{matrix}x\\ y\end{matrix}\right)=\left[\begin{matrix}1&0\\0&1\end{matrix}\right]\left(\begin{matrix}x\\ y\end{matrix}\right)+\left(\begin{matrix}1\\ 0\end{matrix}\right).$ Homework: Section 2.3: Read up to p. 44; #3, 4, 8, 9; Section 2.5: Read all; #1, 2(i)(ii). Also, what affine transformations would you need to create this fractal? Here is some help on using Processing for the first time. Download the Processing code to visualize linear transformations. You will need to copy and paste into a new sketch. Download the Processing code to create fractals using iterated function systems. 
7  Mon, 5 Feb 
Nick will be in Room 315 in the library from 5:00—7:00 tonight. Prof. Tom Banchoff will be speaking at 3:30 in Cowell 107 (and there is food in the Getty Lounge (where the fireplace is in Lo Schiavo) starting at 3:00). Partial solutions to HW: Section 2.3. #3. $PR=\left[\begin{matrix}0&1\\0&0\end{matrix}\right].$ $RP=\left[\begin{matrix}0&0\\1&0\end{matrix}\right].$ #4(a). $\left[\begin{matrix}1&0\\0&0\end{matrix}\right].$ (b). $\left[\begin{matrix}0&0\\0&0\end{matrix}\right].$ (c). $\left[\begin{matrix}19&22\\43&50\end{matrix}\right].$ #8. See 4(b). Partial solutions to HW: Section 2.5. #1. $\left[\begin{matrix}1&11\\3&23\end{matrix}\right].$ Homework problems (assume the cube has as vertices all combinations of $(\pm1,\pm1,\pm1)$ as in class today):

8  Wed, 7 Feb 
Write out all 24 matrices describing the direct symmetries of the cube. Annotate each one with an appropriate picture of a die! Partial homework answers: 1. $\left[\begin{matrix}0&1&0\cr 0&0&1\cr 1&0&0\end{matrix}\right],$ $\left[\begin{matrix}0&0&1\cr 1&0&0\cr 0&1&0\end{matrix}\right],$ and $\left[\begin{matrix}1&0&0\cr 0&1&0\cr 0&0&1\end{matrix}\right].$ 2. $\left[\begin{matrix}0&0&1\cr 0&1&0\cr 1&0&0\end{matrix}\right].$ 
9  Fri, 9 Feb 
Download the book chapter about the hypercube. Get caught up with homework! 
10  Mon, 12 Feb 
Nick's office hours will be in Room 314 in the Library this evening from 5:00—7:00. Homework:

11  Wed, 14 Feb 
Homework answers:
Download solutions to Homework Quiz 1B. In case the first quiz didn't go so well, you can practice with a copy of the version you didn't take. Download a blank copy of Homework Quiz 1A or Homework Quiz 1B. Here are the homework problems on the cross product due Friday. Your assignment is #9, 11, 29, 31, 35, 44, 46, and 47. Here is a summary of the the direct symmetries of the cube. Here is another fractal to practice writing affine transformations which describe its selfsimilarity. 
12  Fri, 16 Feb 

13  Wed, 21 Feb 
Nick's office hours will be in Room 315 in the Library this evening from 5:00—7:00. Download solutions to Homework Quiz 2. Just study for Friday's Midterm! No notes, no calculators, and be on time. NO EXTRA TIME will be given if you arrive late! 
15  Mon, 26 Feb 
Homework: Chapter 2.4, #1—3 and 7—11. Download a blank copy of the first Midterm (Version A) (Version B). Download solutions to the first Midterm. 
16  Wed, 28 Feb 
Nick will be in 315 in the library tonight from 5:00—7:00. Homework answers: 1) $R_{3\pi/4},$ or equivalently, $R_{5\pi/4}.$ 2) $\left[\begin{matrix}1/2 & 0\\0& 1/5\end{matrix}\right]$ 3) $\left[\begin{matrix}1/2 & 1/10\\0& 1/5\end{matrix}\right]$ 7a) The only solution is $(x,y)=(0,0).$ 7b) $\{(x,2x)\,\,x\in{\mathbb R}\}.$ 8a) $(x,y)=(2/5,1/5).$ 8b) No solution — $\{\}$ or $\emptyset.$ 9) $\{(x,12x)\,\,x\in{\mathbb R}\}.$ 10) $\{(x,10x)\,\,x\in\mathbb R\}.$ 11) The system has a solution (in fact, infinitely many solutions) only when $v=5u.$ Homework:

17  Fri, 2 Mar 
HW answers: 2a) $\lambda=1,$ all nonzero vectors are eigenvectors. 2b) $\lambda=1,$ all nonzero vectors are eigenvectors. 2c) $\lambda=0,$ all nonzero vectors are eigenvectors. First matrix: $\lambda=1,$ eigenvectors lie on the line along $\left(\begin{matrix}1\\ 1\\1 \end{matrix}\right).$ Second matrix: $\lambda=1,$ eigenvectors lie on the line along $\left(\begin{matrix}1\\ 1\\1 \end{matrix}\right).$ Third matrix: $\lambda=1,$ all nonzero vectors are eigenvectors. Fourth matrix: $\lambda=1,$ eigenvectors lie on the line along $\left(\begin{matrix}1\\ 0\\1 \end{matrix}\right).$ For $\lambda=1,$ eigenvectors lie on the lines along $\left(\begin{matrix}0\\ 1\\0 \end{matrix}\right)$ and along $\left(\begin{matrix}1\\ 0\\1 \end{matrix}\right).$ Eigenvalues and eigenvectors of $\left[\begin{matrix}1&2\\5&4\end{matrix}\right]:$ Eigenvalue $\lambda=6$ with eigenvector $\left(\begin{matrix}2\\5\end{matrix}\right);$ eigenvalue $\lambda=1$ with eigenvector $\left(\begin{matrix}1\\1\end{matrix}\right).$ Homework from Chapter 2.6: #3(c), #8(c), #9, #10. Also, find the eigenvalues/vectors of the matrix $\left[\begin{matrix}4 & 5\\10 & 1\end{matrix}\right].$ Note: Our book uses $t$ instead of $\lambda,$ which is much more commonly used. Also, I forgot to mention that the equation we use to solve for $\lambda$ is called a characteristic equation. 
18  Mon, 5 Mar 
HW answers: #3(c). $\lambda^21=0$ #8(c). Eigenvalue $\lambda=p$ with eigenvector $\left(\begin{matrix}1\\0\end{matrix}\right);$ eigenvalue $\lambda=q$ with eigenvector $\left(\begin{matrix}0\\1\end{matrix}\right).$ Eigenvalues and eigenvectors of $\left[\begin{matrix}4&5\\10&1\end{matrix}\right]:$ Eigenvalue $\lambda=9$ with eigenvector $\left(\begin{matrix}1\\1\end{matrix}\right);$ eigenvalue $\lambda=6$ with eigenvector $\left(\begin{matrix}1\\2\end{matrix}\right).$ Homework: Chapter 3.6 (p. 170), #8 (assume here that $b\ne0$). Also, find the eigenvalues and eigenvectors of the matrices $\left[\begin{matrix}2&0&0\\0&3&4\\0&4&9\end{matrix}\right]$ (thanks, Wikipedia!) and $\left[\begin{matrix}1&2&2\\2&1&2\\2&2&3\end{matrix}\right]$. Homework Quiz is on Friday this week!!! 
19  Wed, 7 Mar 
Nick will be in Room 315 in the library tonight from 5:00—7:00. Partial answers to the homework: 8(iii): The plane is $z=0.$ First matrix problem: The three eigenvalues are $2,$ $1,$ and $11.$ Corresponding eigenvectors are $\left(\begin{matrix}1\\0\\0\end{matrix}\right),$ $\left(\begin{matrix}0\\2\\1\end{matrix}\right),$ and $\left(\begin{matrix}0\\1\\2\end{matrix}\right).$ Second matrix problem: $\lambda=1,$ with corresponding eigenvectors $\left(\begin{matrix}0\\1\\1\end{matrix}\right)$ and $\left(\begin{matrix}1\\0\\1\end{matrix}\right).$ $\lambda=1,$ with corresponding eigenvector $\left(\begin{matrix}1\\1\\1\end{matrix}\right).$ No homework except to study for Friday's quiz. 
20  Fri, 9 Mar  Have a great Spring Break! 
21  Mon, 19 Mar  Homework: Page 89, Exercises #2, 4, 5, 6. 
22  Wed, 21 Mar 
Nick will be in Room 315 tonight! Homework answers: #2. $D=\left[\begin{matrix}\sqrt{10}&0\\0&\sqrt{10}\end{matrix}\right],$ $P=\left[\begin{matrix}\dfrac{1\sqrt{10}}3&\dfrac{1+\sqrt{10}}3\\1&1\end{matrix}\right]$ #4. $\left[\begin{matrix}128&128\\128&128\end{matrix}\right]$ #5. $\left[\begin{matrix}729&0\\2660&64\end{matrix}\right]$ Solve the following recurrence relation like we did Monday and today in class: $a_{n+2}=5a_{n+1}6a_n,$ $a_0=0,$ $a_1=1.$ 
23  Fri, 23 Mar 
Solution to recurrence relation: $a_n=3^n2^n.$ Homework: p. 131, #19 (note typo: last matrix in (a) should have 1's along the diagonal); p. 143, #20—22. (Note: This HW is moved to Monday.) In addition: #1. Solve the system of equations by finding the LDU decomposition, and then $A^{1}:$ $7x+y=2,$ $y3x=8.$ #2. Solve the system of equations by finding the LDU decomposition, and then $A^{1}:$ $8x6y=16,$ $3x+2y=23.$ Homework answers: #19(a): $\left[\begin{matrix}4&0&0\\4s&5&0\\0&6t&6\end{matrix}\right]$ #19(b): $\left[\begin{matrix}0&0&2\\0&0&0\\0&2&0\end{matrix}\right]$ Homework answers: #1. $L,$ $D,$ $U,$ and $A^{1}$ are: $\left[\begin{matrix}1&0\\3/7&1\end{matrix}\right],$ $\left[\begin{matrix}7&0\\0&10/7\end{matrix}\right],$ $\left[\begin{matrix}1&1/7\\0&1\end{matrix}\right],$ and $\left[\begin{matrix}1/10&1/10\\3/10&7/10\end{matrix}\right].$ $(x,y)=(1,5).$ #2. $L,$ $D,$ $U,$ and $A^{1}$ are: $\left[\begin{matrix}1&0\\3/8&1\end{matrix}\right],$ $\left[\begin{matrix}8&0\\0&17/4\end{matrix}\right],$ $\left[\begin{matrix}1&3/4\\0&1\end{matrix}\right],$ and $\left[\begin{matrix}1/17&3/17\\3/34&4/17\end{matrix}\right].$ $(x,y)=(5,4).$ Reminder: Homework quiz next Wednesday! 
24  Mon, 26 Mar 
Download solutions to Homework Quiz 3. Homework: p. 131, #19 (note typo: last matrix in (a) should have 1's along the diagonal); p. 143, #20—22. Also: #1. Show that the eigenvectors of $\left[\begin{matrix}1&4\\4&7\end{matrix}\right]$ are orthogonal. #2. Show that the eigenvectors of $\left[\begin{matrix}3&12\\12&7\end{matrix}\right]$ are orthogonal. Homework Quiz is on Wednesday! 
25  Wed, 28 Mar 
Homework answers (see Day 23 for answers to problems from the book): #1. $\lambda=9$ has eigenvector $\left(\begin{matrix}1\\2\end{matrix}\right),$ $\lambda=1$ has eigenvector $\left(\begin{matrix}2\\1\end{matrix}\right).$ #2. $\lambda=15$ has eigenvector $\left(\begin{matrix}2\\3\end{matrix}\right),$ $\lambda=11$ has eigenvector $\left(\begin{matrix}3\\2\end{matrix}\right).$ Enjoy the long weekend! 
26  Mon, 2 Apr 
Homework: Solve the following system of equations by finding an LDU decomposition of the appropriate matrix. The system is You should have $L=\left[\begin{matrix}1&0&0\\1/2&1&0\\2&2/3&1\end{matrix}\right],\quad D=\left[\begin{matrix}2&0&0\\0&3/2&0\\0&0&3\end{matrix}\right],\quad U=\left[\begin{matrix}1&1/2&1/2\\0&1&1\\0&0&1\end{matrix}\right].$ The solution is $(x,y,z)=(3,2,1).$ Here are some additional practice problems: #1. Find the $LDU$ decomposition of $\left[\begin{matrix}5&2\\3&1\end{matrix}\right].$ #2. Solve the recurrence $a_{n+2}=8a_{n+1}15a_n,\qquad a_0=0,\quad a_1=2.$ Answers: #1. $L,$ $D,$ and $U,$ are: $\left[\begin{matrix}1&0\\3/5&1\end{matrix}\right],$ $\left[\begin{matrix}5&0\\0&1/5\end{matrix}\right],$ and $\left[\begin{matrix}1&2/5\\0&1\end{matrix}\right].$ #2. $a_n=3^n5^n.$ Don't forget the Midterm on Friday! 
27  Wed, 4 Apr 
Nick has office hours tonight in the library. Download solutions to Homework Quiz 4. Midterm on Friday! 
29  Mon, 9 Apr  Download solutions to Midterm 2. 
30  Wed, 11 Apr  Link to Homework: #1, 6, 9, 10, 12, 15, 16, 18, 25, 28. 
31  Fri, 13 Apr 
Homework Quiz next Friday! Practice counting poker hands with Durango Bill. Be able to count the first seven examples of numbers of fivecard poker hands if there are no wild cards. Several explanations are given on the web page. 
32  Mon, 16 Apr 
Here is a link to your open source probability textbook. Don't forget to note on the site about solutions to the oddnumbered prioblems! Read Section 1.2. Homework (starting on p. 35): #1, 4—9. Don't forget your Homework Quiz on Friday! 
33  Wed, 18 Apr 
Answers to p. 35: #4 (a) first toss is H; (b) all the same toss; (c) exactly one tail; (d) at least one tail. #6. A two has $2/21$ chance, a four $4/21$ chance, and a six has a $6/21$ chance. Thus, the probability of rolling an even number is $12/21=4/7.$ #8. Art and psychology are $1/4,$ and geology is $2/4=1/2.$ Here is the link to the generating functions worksheet. Use this worksheet for your generating functions homework. NOTE: I did #4 in class today by mistake. So instead, do the problem with 5cent and 8cent coins. 
34  Fri, 20 Apr  Finish the homework assignment on generating functions! 
35  Mon, 23 Apr 
NOTE: The Final Exam for the 9:15 class is scheduled for Wednesday, 16 May, from 10:00—1:00, and the Final Exam for the 10:30 class is scheduled for Monday, 14 May, from 10:00—1:00. You must take your exam on the corresponding date! Changing exam times requires written permission from the Dean's office. The review session for the Final Exam is tentatively scheduled for Friday, 11 May, from 10:00—11:30. Download last year's Final Exam. Download solutions to last year's Final Exam. Homework: p. 71, #1, 7, 8. 
36  Wed, 25 Apr 
Homework Quiz Friday. Last one! Homework (for Monday): p. 73, #12 (I'm not sure the hint given really helps), 15. Also, the following problem: Suppose a circle of diameter 10 cm is inscribed in a square of side length 10 cm. If a coin of radius 1 cm is tossed so that it lies entirely within the square, what is the probability that it lies entirely within the circle? We will also go over the generating functions homework on Friday, so at least try to do a few of these before then! 
37  Fri, 27 Apr  Finish the homework on geometric probability assigned on Wednesday. 
38  Mon, 30 Apr 
Download solutions to Homework Quiz 5. The Review Session on Friday, May 11 from 10:00—11:30 is in Harney 136, not our normal classroom! Homework Problem 1: Suppose you and your friend arrived at Fisherman's Wharf last Saturday sometime between 1:00 and 8:00. You stayed exactly one hour, and your friend stayed exactly three hours. Neither of you knew that the other was there. Assuming a uniformly distributed time of arrival, what is the probability that at some time, you were both at Fisherman's Wharf last Saturday? Inclass Problem: A CD is 60 minutes long. An 8minute conversation is recorded on the CD. You accidentally erase 15 consecutive minutes of the CD. Find the probability that:

39  Wed, 2 May  Homework: #2, 3, and 5 on page 150. 
41  Mon, 7 May  Download solutions to Homework Quiz 6. 
42  Wed, 9 May  Download blank copies of Midterm 2A and Midterm 2B. 