Linear Algebra and Probability, Spring 2016 
1  Mon, 25 Jan  Download the Course Syllabus.
Homework: Read pp. 3—11, and do Exercises #1—5, 7. 
2  Wed, 27 Jan  Homework: Finish reading Section 2.0 (but skip all the trigonometric derivations on pp. 14—16). Do problems #4, 5 (using parametric equations), 9—10, 12—16, and 18. 
3  Fri, 29 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! 
4  Mon, 1 Feb 
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. $\left[\begin{matrix}1&0\\0&1\end{matrix}\right].$ Today's homework:

5  Wed, 3 Feb 
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 $g\circ f\left(\begin{matrix}x\\ y\end{matrix}\right)\left(\begin{matrix}y\\ x\end{matrix}\right).$ Today's homework:
Visit the Sage page with fractals generated by affine transformations. Note: To change any of these fractals, you will need to sign up for a free account and copy the page to a new project of your own. To execute a function, put your cursor somewhere in the middle of it and then hit Shift+Enter (in other words, hit Enter while you are already holding down the Shift key). Finally, here is a link to the Spring 2015 website. Homework quizzes, sample exams, and much, much more! 
6  Fri, 5 Feb 
Answers to yesterday's homework: 2. $f\left(\begin{matrix}x\\ y\end{matrix}\right)=\left[\begin{matrix}1&0\\0&1\end{matrix}\right]\left(\begin{matrix}x\\ y\end{matrix}\right)+\left(\begin{matrix}3\\ 2\end{matrix}\right).$ 3. $\left[\begin{matrix}0&3\\2&0\end{matrix}\right].$ 4. $f\left(\begin{matrix}x\\ y\end{matrix}\right)=\left[\begin{matrix}1&0\\0&1\end{matrix}\right]\left(\begin{matrix}x\\ y\end{matrix}\right)+\left(\begin{matrix}1\\ 0\end{matrix}\right).$ Download the prompt for your fractal project. To help with this project, you may download the $\LaTeX$ code which produced the template. Then download $\LaTeX$ on your computer, go to www.sharelatex.com, or simply create a $\LaTeX$ document in the Sage environment. Also, there are some resources at Art of Problem Solving under the Resources menu. This is not a long assignment in terms of the number of pages, but there's a lot to learn to complete the assignment (which is why you have until the 19th). Feel free to use and adapt the code on my Sage worksheet — but you should know that you cannot earn an A just by changing the numbers in my code. In order to earn an A, I must see you either use some significant features of Sage in a way not illustrated in the worksheet, or include significantly different Python code within your worksheet. Homework: Section 2.3: Read up to p. 44; #3, 4, 8, 9; Section 2.5: Read all; #1, 2(i)(ii). 
7  Mon, 8 Feb 
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):
Don't forget the HW quiz Wednesday! 
8  Wed, 10 Feb 
Finish whatever Homework you didn't finish already. In addition, 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, 12 Feb  WORK ON YOUR PROJECT!!!! There will be NO EXTENSIONS GIVEN. 
10  Wed, 17 Feb  Homework:

11  Fri, 19 Feb 
Homework answers:
Download the first Midterm from Spring 2015. 
12  Mon, 22 Feb 
Answers to cross product questions:
Note the correction to #3 on Day 6. 
13  Wed, 24 Feb 
Download solutions to Homework Quiz 2. Exam Friday! 
14  Fri, 26 Feb  First Exam 
15  Mon, 29 Feb  Homework: Chapter 2.4, #1—3 and 7—11. 
16  Wed, 2 Mar 
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: #2 on p. 76. Also, take out your die and find all eigenvalues and eigenvectors for the four matrices listed on Day 8 (10 Feb). You should be able to do this geometrically with your knowledge of threedimensional coordinates! 
17  Fri, 4 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 $\left(\begin{matrix}1\\ 0\\1 \end{matrix}\right).$ 
18  Mon, 7 Mar  HW from Chapter 2.6: #3(a)(c)(d), #8(c)(d), #9, #12. Also, find all eigenvalues and eigenvectors of $\left[\begin{matrix}1&2\\5&4\end{matrix}\right].$ 
19  Wed, 9 Mar 
HW answers: #3(a). $(\lambdar)^2=0$ #3(c). $\lambda^21=0$ #3(d). $\lambda^21=0$ #8(c). Eigenvalue $\lambda=p$ with eigenvector $\left(\begin{matrix}1\\0\end{matrix}\right);$ eigenvalue $\lambda=q$ with eigenvector $\left(\begin{matrix}0\\1\end{matrix}\right).$ #8(d). Eigenvalue $\lambda=0$ with eigenvector $\left(\begin{matrix}1\\0\end{matrix}\right).$ 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: Chapter 3.6 (p. 170), #7, #8 (assume here that $b\ne0$). Also, find the eigenvalues and eigenvectors of the matrix $\left[\begin{matrix}2&0&0\\0&3&4\\0&4&9\end{matrix}\right]$ (thanks, Wikipedia!). Download solutions to the first Midterm. Homework Quiz Friday!!! 
20  Fri, 11 Mar 
Homework answers: #7. The three eigenvalues are $\sqrt2,$ $\sqrt2,$ and $1.$ Corresponding eigenvectors are $\left(\begin{matrix}\sqrt2\\0\\1\end{matrix}\right),$ $\left(\begin{matrix}\sqrt2\\0\\1\end{matrix}\right),$ and $\left(\begin{matrix}0\\1\\0\end{matrix}\right).$ #8. The plane is $z=0.$ Third 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 $\left(\begin{matrix}0\\1\\2\end{matrix}\right).$ 
21  Mon, 21 Mar 
Download solutions to Homework Quiz 3. Homework: Page 89, Exercises #4, 5, 6, 2. 
22  Wed, 23 Mar 
Homework answers: #4. $\left[\begin{matrix}128&128\\128&128\end{matrix}\right]$ #5. $\left[\begin{matrix}729&0\\2660&64\end{matrix}\right]$ #2. $D=\left[\begin{matrix}\sqrt{10}&0\\0&\sqrt{10}\end{matrix}\right],$ $P=\left[\begin{matrix}\dfrac{1\sqrt{10}}3&\dfrac{1+\sqrt{10}}3\\1&1\end{matrix}\right]$ Homework: p. 131, #19 (note typo: last matrix in (a) should have 1's along the diagonal); p. 143, #20—22. Also, solve the following recurrence relation like we did Monday in class: $a_{n+2}=5a_{n+1}6a_n,$ $a_0=0,$ $a_1=1.$ Homework answers: #19(a): $\left[\begin{matrix}4&0&0\\4s&5&0\\0&6t&6\end{matrix}\right]$ #19(b): $\left[\begin{matrix}0&0&2\\0&0&0\\0&2&0\end{matrix}\right]$ Recurrence relation: $a_n=3^n2^n.$ Download the demo movie we discussed in class on March 11. You will be making a similar movie for your project. The prompt is essentially like the prompt for the first project: create a movie morphing one fractal into another. You can add more transformations, change background colors, represent different transformations with different colors, etc. If all you do is change the numbers in my code, you will NOT earn a passing grade! I need to see that you've put some thought into the project! Download a blank copy of the second Midterm form Spring 2015. Note that the coverage of topics is NOT the same as Spring 2015. This semester's exam will ONLY be based on the HW since the first Midterm. Download a summary of all direct symmetries of the cube. 
23  Mon, 28 Mar 
#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 $A^{1}:$ $8x6y=16,$ $3x+2y=23.$ Don't forget the HW quiz on Wednesday! 
24  Wed, 30 Mar 
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 $\left[\begin{matrix}1/17&3/17\\3/34&4/17\end{matrix}\right].$ $(x,y)=(5,4).$ Homework: #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. REMINDER: The Final Exam is on Wednesday, May 18 from 12:30—3:30 in KA 267. 
25  Fri, 1 Apr 
Homework answers: #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 $\left(\begin{matrix}3\\2\end{matrix}\right).$ Download last year's Second Midterm. Do problems #7, 9, 11, 12, 13, 18, 19, 20. Answers are on Day 32 of the Spring 2015 website. NOTE: Office Hours are CANCELLED Monday!!! 
26  Mon, 4 Apr  Download solutions to Homework Quiz 4. 
27  Wed, 6 Apr 
Practice problems: #1. Find the $LDU$ decomposition of $\left[\begin{matrix}5&2\\3&1\end{matrix}\right].$ #2. Find the $LDU$ decompoistion of $\left[\begin{matrix}1&2\\4&6\end{matrix}\right].$ #3. Solve the recurrence $a_{n+1}=8a_{n+1}15a_n,\qquad a_0=0,\quad a_1=2.$ 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. $L,$ $D,$ and $U,$ are: $\left[\begin{matrix}1&0\\4&1\end{matrix}\right],$ $\left[\begin{matrix}1&0\\0&14\end{matrix}\right],$ and $\left[\begin{matrix}1&2\\0&1\end{matrix}\right].$ #3. $a_n=3^n5^n.$ 
28  Fri, 8 Apr  Second Exam 
29  Mon, 11 Apr  Last day to drop courses! Remember, your Homework is to learn all about the fivecard poker hands (with NO Jokers or wild cards), and to learn how to count how many full houses there are. Download solutions to the second Midterm exam. 
30  Wed, 13 Apr 
Homework: Count the number of the following poker hands.
Answers: (1) 502,860. (2) 137,280. 
31  Fri, 15 Apr 
Link to Homework: #1, 6, 9, 10, 12, 15, 16, 18, 25, 28. Homework Quiz next Wednesday! Practice counting poker hands with Durango Bill. 
32  Mon, 18 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 Wednesday! 
33  Wed, 20 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 $2/4=1/2.$ Download the prompt for the fractal movie project. Here is the link to the generating functions worksheet. Use this worksheet for your generating functions homework. Finally, make sure you download Processing and get my file to load. We'll talk more on Friday  but don't put this off. At least get something working! 
34  Fri, 22 Apr  Homework: p. 71, #1, 7, 8. 
35  Mon, 25 Apr  No new homework. Get caught up! We'll be going over the past several days' homework on Wednesday. 
36  Wed, 27 Apr 
Homework Quiz on Friday. Last one! Download solutions to Homework Quiz 5. Answers to generating functions questions not covered in class: 1) Same as for normal dice! 2) Even is $4/9,$ odd is $5/9.$ 
37  Fri, 29 Apr  Project IN MY HAND BY 3:15 MONDAY!!!!! 
38  Mon, 2 May 
Homework: p. 73, #12, 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? Additional problem: Suppose you and your friend arrived at Fisherman's Wharf last Saturday sometime between 1:00 and 5:00. You stayed exactly one hour, and your friend stayed exactly two 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? 
39  Wed, 4 May 
Homework 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:

40  Fri, 6 May 
Don't forget the Review Session for the Final Exam next Friday at 10:00 in our usual classroom! Download solutions to the last Homework Quiz. Download a blank copy of the First Midterm. Download a blank copy of the Second Midterm. Download a blank copy of the Spring 2015 Final Exam. Questions from last year's Final Exam which will NOT be covered on this year's exam are #1, 5, 14, 16. 
41  Mon, 9 May  Download solutions to the practice Final Exam. 
42  Wed, 11 May 
Download a blank copy of Homework Quiz 1. Download a blank copy of Homework Quiz 2. Download a blank copy of Homework Quiz 3. Download a blank copy of Homework Quiz 4. Download a blank copy of Homework Quiz 5. Download a blank copy of Homework Quiz 6. 