Linear Algebra and Probability, Spring 2016

Week 1

 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!

Week 2

 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). $x'=\dfrac{x-y}2,\quad y'=\dfrac{y-x}2.$ 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: Write the following affine transformation using its linear part and its translation, and describe its effect on a unit square: $f\left(\begin{matrix}x\\ y\end{matrix}\right)=\left(\begin{matrix}-2x-2\\y+1\end{matrix}\right).$ Also, rewrite $f$ using matrix notation. Same instructions as #1: $f\left(\begin{matrix}x\\ y\end{matrix}\right)=\left(\begin{matrix}x+1\\-y\end{matrix}\right).$ This is an example of a glide reflection. Google it! Suppose that $A$ is the linear transformation which rotates $90^\circ$ counterclockwise, and that $B$ is the linear transformation which reflects across the $y$-axis. Now start with a unit square (with arrows appropriately drawn). On one figure, first do $A,$ and then do $B$ to this result. On a second figure, first do $B,$ and then do $A$ to this result. You should have obtained different figures. This means that, in general, performing linear transformations is not a commutative operation. A common illustration of this fact is having $A$ be the process of putting on your socks, and $B$ be the process of putting on your shoes. Order matters! However, in some cases, the order doesn't matter. Can you find two linear transformations for which order does not matter? Let $f\left(\begin{matrix}x\\ y\end{matrix}\right)=\left(\begin{matrix}x/2+1/2\\ y/2\end{matrix}\right)$ and $g\left(\begin{matrix}x\\ y\end{matrix}\right)=\left(\begin{matrix}-y/2+1\\ x/2\end{matrix}\right).$ Describe the effect of each of these transformations. You should have found that they both transform the unit square to the same position, that the square is oriented differently (the arrows are in different places). This is a significant difference, as we will see later. Let $f\left(\begin{matrix}x\\ y\end{matrix}\right)=\left(\begin{matrix}-x\\y\end{matrix}\right)$ and $g\left(\begin{matrix}x\\ y\end{matrix}\right)=\left(\begin{matrix}y\\x\end{matrix}\right).$ Look up the concept of function composition, which you likely learned sometime before, but maybe forgot. Evaluate $f\circ g\left(\begin{matrix}x\\ y\end{matrix}\right)$ and $g\circ f\left(\begin{matrix}x\\ y\end{matrix}\right).$ 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: We mentioned in class today that a rotation and a scaling commute, that is, performing these transformations in either order produces the same result. However, is this the case if the $x$ and $y$ directions are scaled by different amounts? Justify your result! You may use a specific example if you like. Write the affine transformation which first rotates by $180^\circ,$ and then moves to the right $3$ units and down $2$ units. Write the linear transformation which first scales the $x$ by a factor of $2,$ then scales the $y$ by a factor of $3,$ and then rotates $90^\circ$ clockwise. Order matters! (Hint: Use function composition, which is really the same thing as matrix multiplication.) (Trickier!) Write the affine transformation which moves to the right $2$ units, then reflects about the $y$-axis, and then moves to the right $1$ unit. Look at this fractal of the P-pentomino, one of my favorite fractal images. What affine transformations would you use to create this fractal? Thinking about this will greatly help you with the writing project! 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).

Week 3

 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): You are holding the cube by the opposite corners $(1,1,1)$ and $(-1,-1,-1)$. You spin it one-third the way around, then one-third again, and the third time you're back to where you started. What $3\times3$ matrices describe these rotations? You are holding the cube by the midpoint of the edge with endpoints $(1,1,1)$ and $(1,-1,1)$, and by the midpoint of the opposite edge. You spin the cube $180^\circ$ around the axis through these two midpoints. What matrix describes this rotation? Imagine the cube as drawn in class. Let $A$ be the matrix describing the rotation bringing the top toward you $90^\circ$ (the axis of rotation is the $y$-axis here). Let $B$ be the matrix describing the rotation which turns the cube $90^\circ$ to your right (the axis of rotation is the $x$-axis). By sheer force of imagination, write the matrix describing what happens when the rotation $B$ is done first, and then the rotation $A$ is performed. Then, compute the matrix product $AB$ to see that your answer is correct. What rotation corresponds to the matrix $\left[\begin{matrix}0&-1&0\cr 0&0&1\cr -1&0&0\end{matrix}\right]$? Hey, wait minute! You've been writing out tons of matrices, and they all seem to have two zeroes in each row and column — and the other element is either $1$ or $-1.$ Surely this is no concidence? Explain why this is so by giving a geometrical interpretation of the effect of these matrices on the cube. How many direct symmetries of the cube are there? How many opposite symmetries? Why? How many direct symmetries of the hypercube are there? Opposite symmetries? Download solutions to Homework Quiz 1. 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.

Week 4

 10 Wed, 17 Feb Homework: Find the angle between $\left(\begin{matrix}1\\ -1\\0 \end{matrix}\right)$ and $\left(\begin{matrix}2\\ 1\\-1\end{matrix}\right).$ Find the projection of $\left(\begin{matrix}-3\\ 1\\4\end{matrix}\right)$ onto the vector $\left(\begin{matrix}1\\ 2\\-2\end{matrix}\right).$ Find the projection of $\left(\begin{matrix}-3\\ 1\\4\end{matrix}\right)$ onto the plane $-2x+y-4z=0.$ Find the distance from the point $\left(\begin{matrix}2\\ 0\\3\end{matrix}\right)$ to the plane $-2x+y-4z=0.$ Find the distance from the point $\left(\begin{matrix}-2\\ 4\\3\end{matrix}\right)$ to the line through the origin parallel to $\left(\begin{matrix}4\\ 0\\-3 \end{matrix}\right).$ And don't forget that your projects are due Friday!!!!! 11 Fri, 19 Feb Homework answers: To two decimal places, the angle is $73.22^\circ.$ $\left(\begin{matrix}-1\\ -2\\2 \end{matrix}\right).$ $\left(\begin{matrix}-27/7\\ 10/7\\16/7 \end{matrix}\right).$ $16/\sqrt{21}.$ $2\sqrt{109}/5.$ Here are the Homework problems on the cross product due Monday. Your assignment is #9, 11, 29, 31, 35, 44, 46, and 47. Download the first Midterm from Spring 2015.

Week 5

 12 Mon, 22 Feb Answers to cross product questions: #9(a). $\left(\begin{matrix}17\\ -33\\-10 \end{matrix}\right)$ (b). $\left(\begin{matrix}-17\\ 33\\10 \end{matrix}\right)$ (c). $\left(\begin{matrix}0\\ 0\\0 \end{matrix}\right)$ #11. $\left(\begin{matrix}-1\\ -1\\-1 \end{matrix}\right)$ #29. $6\sqrt5$ #31. $2\sqrt{83}$ #35. $\dfrac{\sqrt{16742}}2$ #44. $0$ #46. $-72$ #47. $75$ The only homework is to start studying for your exam on Friday! Bring questions to the in-class review on Wednesday, and visit DK Wednesday evening from 6—8:00 in the 5th floor computer lab. 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

Week 6

 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) $\{(x,1-2x)\,|\,x\in{\mathbb R}\}.$ 10) $\{(x,10-x)\,|\,x\in\mathbb R\}.$ 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 three-dimensional 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 $\left(\begin{matrix}1\\ 0\\1 \end{matrix}\right).$

Week 7

 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). $(\lambda-r)^2=0$ #3(c). $\lambda^2-1=0$ #3(d). $\lambda^2-1=0$ #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 $\left(\begin{matrix}0\\1\\2\end{matrix}\right).$

Week 8

 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: $a_n=3^n-2^n.$ 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.

Week 9

 23 Mon, 28 Mar #1. Solve the system of equations by finding the LDU decomposition, and then $A^{-1}:$ $7x+y=-2,$ $y-3x=8.$ #2. Solve the system of equations by finding the LDU decomposition, and then $A^{-1}:$ $8x-6y=16,$ $3x+2y=23.$ 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!!!

Week 10

 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. $a_n=3^n-5^n.$ 28 Fri, 8 Apr Second Exam

Week 11

 29 Mon, 11 Apr Last day to drop courses! Remember, your Homework is to learn all about the five-card 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. Ace high (that is, there are no pairs or higher hand, and the highest card is an ace). This is a little tricky! There are many straights (AKQJ10 and A2345) and flushes with five different cards and an ace. So first, calculate the number of all hands with five different cards with an ace. Then subtract: the number of straight flushes (include both types of straights); the number of AKQJ10 straights which are not also flushes; the number of A2345 straights which are not also flushes; the number of flushes with an ace which are not also straights. Note: There cannot be a Joker here, since that would automatically make a pair! Three-of-a-kind hands. Assume a 53-card deck (so your deck has a Joker in it). 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.

Week 12

 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 odd-numbered 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.

Week 13

 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!!!!!

Week 14

Week 15