Linear Algebra and Probability, Spring 2017 
1  Mon, 23 Jan  Download the Course Syllabus.
Homework: Read pp. 3—11, and do Exercises #1—5, 7. 
2  Wed, 25 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. Don't forget Ethan's regular Thursday office hour from 11:50—12:50 in the CS lab on the 5^{th} floor of Harney! 
3  Fri, 27 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! Feel free to browse last year's course website for sample quizzes, exams, and much, much more!! 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, 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. $\left[\begin{matrix}1&0\\0&1\end{matrix}\right].$ Here is a link to today's Sage worksheet on affine transformations. Today's homework:

5  Wed, 1 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). Here is a helpful summary of affine transformations. 
6  Fri, 3 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. $f\left(\begin{matrix}x\\ y\end{matrix}\right)=\left[\begin{matrix}1&0\\0&1\end{matrix}\right]\left(\begin{matrix}x\\ y\end{matrix}\right)+\left(\begin{matrix}1\\ 0\end{matrix}\right).$ Download the prompt for your fractal project. To help with this project, you may download the $\LaTeX$ code (and the fractal image in the document) 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 17th). 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, 6 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):

8  Wed, 8 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, 10 Feb 
Download solutions to Homework Quiz 1. Download a blank copy of Homework Quiz 1. Download a helpful list of rotation matrices in three dimensions. WORK ON YOUR PROJECT!!!! There will be NO EXTENSIONS GIVEN. 
10  Wed, 17 Feb  Homework:
And don't forget that your projects are due Friday!!!!! In light of this fact, I have decided that there is no homework quiz this week! 
11  Fri, 19 Feb 
Homework answers:
Don't forget Ethan's office hour tomorrow in the 5th floor lab at 11:40 if you need help with your project due Friday! Additional note for your projects: each choice of random number should have exactly one transformation associated with it. Otherwise, you are not using the correct algorithm! Also, please upload the link to your Sage document on Canvas. I just created an assignment for this! Remember, if you need to email me, DO NOT send a Canvas email, as I do not read it! 
12  Fri, 17 Feb 
Download solutions to the Homework Quiz 2. Answers to cross product questions:
Remember: Upload a link to your Sage worksheet on Canvas! You must do this even if you have already emailed me your link! You have until midnight tonight if you have not already done so. Additional practice problems for the Midterm:

13  Wed, 22 Feb 
Just study for your Midterm on Friday! Remember, no notes or calculator are allowed, but you may bring your die with you. Remember also that Ethan is available Thursday at 11:40 in the 5th floor lab to answer questions! 
14  Fri, 24 Feb  First Exam 
15  Mon, 27 Feb 
Homework: Chapter 2.4, #1—3 and 7—11. Download a blank copy of the first Midterm. Download solutions to the first Midterm. 
16  Wed, 1 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:

17  Fri, 3 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 $\left(\begin{matrix}1\\1\end{matrix}\right).$ Download the demo movie we discussed in class today. 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! This project is due Monday, April 10. This is the Monday after the second Midterm, so plan ahead! It's not a good idea to wait until the last minute (like with the last project). You should be able to find ample time around your CS projects to get this done. Download the complete prompt ($\LaTeX$ code). Homework from Chapter 2.6: #3(a)(c)(d), #8(c)(d), #9, #10. 
18  Mon, 6 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), #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 moved to Friday!!! 
19  Wed, 8 Mar 
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, 10 Mar  Have a great Spring Break! 
21  Mon, 20 Mar 
Homework: Page 89, Exercises #2, 4, 5, 6. REMINDER: Office Hours ONLY from 10:40—11:40 tomorrow due to a department meeting! 
22  Wed, 22 Mar 
Download solutions to Homework Quiz 3. 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. $\left[\begin{matrix}729&0\\2660&64\end{matrix}\right]$ 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: $a_n=3^n2^n.$ 
23  Fri, 24 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 next Wednesday! Additional practice with recurrence relations: $a_{n+2}=4a_{n+1}3a_{n}$, $a_0=1,$ $a_1=5.$ 
24  Mon, 27 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 for the 9:15 section is on Wednesday, May 17 from 10:00—1:00 in CO 313. The Final Exam for the 10:30 section is on Monday, May 15 from 10:00—1:00 in CO 313. You must take the exam in the time slot for your assigned section. Written permission from the Dean is required to change the time of your Final Exam. 
25  Wed, 29 Mar 
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 $\left(\begin{matrix}3\\2\end{matrix}\right).$ 
26  Fri, 31 Mar 
Solve the following system of equations. First, find an LDU decomposition of the appropriate matrix. Then use it to find and inverse matrix, and check this by using the formula we learned in class.
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 $(x,y,z)=(3,2,1).$ 
27  Mon, 3 Apr 
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. $a_n=3^n5^n.$ Come prepared for the review on Wednesday! 
28  Wed, 5 Apr 
Download solutions to Homework Quiz 4. Projects are due next Monday! Bring a hard copy to class to hand in. Remember to bring your die if you want it for Friday! 
29  Fri, 7 Apr  Second Exam. REMINDER: Upload your movie (go to the Tools menu, and choose Movie Maker) and your .pyde file to Canvas by Monday at midnight. Hand in your hard copy at the beginning of class. And make sure you know the cards of a standard 52card deck since we'll be counting on Monday.... 
30  Mon, 10 Apr  Make sure you upload the appropriate items before midnight tonight. Download solutions to the second Midterm exam. Last day to drop courses! Link to Homework: #1, 6, 9, 10, 12, 15, 16, 18, 25, 28. 
31  Wed, 12 Apr 
Homework Quiz next Friday! Practice counting poker hands with Durango Bill. Be able to count the number of fivecard poker hands if there are no wild cards. Several explanations are given on the web page. 
32  Mon, 17 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, 19 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. Download last year's Final Exam. 
34  Fri, 21 Apr  Finish the generating functions homework! 
35  Mon, 24 Apr 
Homework: p. 71, #1, 7, 8.
Download solutions to the Spring 2016 Final Exam. 
36  Wed, 26 Apr 
Homework Quiz next Monday. Last one! 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? 
37  Fri, 28 Apr 
No homework except to study for the Homework Quiz on Monday! Reminder: There is a review session for the Final Exam on Friday, May 12, from 11:00—1:00 in our usual classroom. 