Linear Algebra and Probability, Spring 2018


Week 1

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. $x=3-2t, y=1+t$ (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. $\sqrt5.$


Week 2

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). $x'=\dfrac{x-y}2,\quad y'=\dfrac{y-x}2.$

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. $\left[\begin{matrix}1&0\\0&1\end{matrix}\right].$

Download a useful summary of linear and affine transformations.

Today's homework:
  1. Write the following affine transformation using its linear part and its translation, and describe its effect on a unit square: $f\left(\begin{matrix}x\\ y\end{matrix}\right)=\left(\begin{matrix}-2x-2\\y+1\end{matrix}\right).$
  2. Also, rewrite $f$ using matrix notation.

  3. 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!


  4. 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!


  5. However, in some cases, the order doesn't matter. Can you find two linear transformations for which order does not matter?


  6. 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.


  7. Let $f\left(\begin{matrix}x\\ y\end{matrix}\right)=\left(\begin{matrix}-x\\y\end{matrix}\right)$ and $g\left(\begin{matrix}x\\ y\end{matrix}\right)=\left(\begin{matrix}y\\x\end{matrix}\right).$
  8. 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, 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:
  1. Write the affine transformation which first rotates by $180^\circ,$ and then moves to the right $3$ units and down $2$ units.

  2. 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.)

  3. (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.

  4. Look at this fractal of the L-tromino, one of my favorite fractal images. What affine transformations would you use to create this fractal?
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.


Week 3

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):

  1. 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?

  2. 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?

  3. 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.

  4. What rotation corresponds to the matrix $\left[\begin{matrix}0&-1&0\cr 0&0&1\cr -1&0&0\end{matrix}\right]$?

  5. 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.

  6. How many direct symmetries of the cube are there? How many opposite symmetries? Why?

  7. How many direct symmetries of the hypercube are there? Opposite symmetries?
Don't forget the HW quiz Wednesday!
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!


Week 4

10 Mon, 12 Feb Nick's office hours will be in Room 314 in the Library this evening from 5:00—7:00.

Homework:
  1. Read Chapter 3.0 (up to p. 106).


  2. Find the angle between $\left(\begin{matrix}1\\ -1\\0 \end{matrix}\right)$ and $\left(\begin{matrix}2\\ 1\\-1\end{matrix}\right).$

  3. Find the projection of $\left(\begin{matrix}-3\\ 1\\4\end{matrix}\right)$ onto the vector $\left(\begin{matrix}1\\ 2\\-2\end{matrix}\right).$

  4. Find the projection of $\left(\begin{matrix}-3\\ 1\\4\end{matrix}\right)$ onto the plane $-2x+y-4z=0.$

  5. Find the distance from the point $\left(\begin{matrix}2\\ 0\\3\end{matrix}\right)$ to the plane $-2x+y-4z=0.$

  6. Find the distance from the point $\left(\begin{matrix}-2\\ 4\\3\end{matrix}\right)$ to the line through the origin parallel to $\left(\begin{matrix}4\\ 0\\-3 \end{matrix}\right).$

11 Wed, 14 Feb Homework answers:
  1. No answer needed....

  2. To two decimal places, the angle is $73.22^\circ.$

  3. $\left(\begin{matrix}-1\\ -2\\2 \end{matrix}\right).$

  4. $\left(\begin{matrix}-27/7\\ 10/7\\16/7 \end{matrix}\right).$

  5. $16/\sqrt{21}.$

  6. $2\sqrt{109}/5.$
Download solutions to Homework Quiz 1A.

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 self-similarity.
12 Fri, 16 Feb
  1. #9(a). $\left(\begin{matrix}17\\ -33\\-10 \end{matrix}\right)$ (b). $\left(\begin{matrix}-17\\ 33\\10 \end{matrix}\right)$ (c). $\left(\begin{matrix}0\\ 0\\0 \end{matrix}\right)$

  2. #11. $\left(\begin{matrix}-1\\ -1\\-1 \end{matrix}\right)$

  3. #29. $6\sqrt5$

  4. #31. $2\sqrt{83}$

  5. #35. $\dfrac{\sqrt{16742}}2$

  6. #44. $0$

  7. #46. $-72$

  8. #47. $75$

Additional practice problems for the Midterm:
  1. Find both parametric and symmetric equations for the line through the points $\left(\begin{matrix}0\\ 3\\-1 \end{matrix}\right)$ and $\left(\begin{matrix}2\\ 4\\3 \end{matrix}\right).$
  2. Find symmetric equations for the line through the points $\left(\begin{matrix}0\\ 3\\-1 \end{matrix}\right)$ and $\left(\begin{matrix}2\\ 3\\3 \end{matrix}\right).$
  3. Find an equation of the plane with normal $\left(\begin{matrix}-6\\ 3\\1 \end{matrix}\right)$ which passes through the point $\left(\begin{matrix}2\\ 0\\3 \end{matrix}\right).$
  4. Find an equation for the plane passing through $\left(\begin{matrix}4\\ 1\\0 \end{matrix}\right),$ $\left(\begin{matrix}3\\ 0\\1 \end{matrix}\right),$ and $\left(\begin{matrix}-1\\ -5\\3 \end{matrix}\right).$
Answers:
  1. Parametric equations — there are several correct answers, here is one: $x=2t,$ $y=3+t,$ $z=-1+4t.$

    Symmetric equations: $\dfrac x2=y-3=\dfrac{z+1}4.$
  2. $\dfrac x2=\dfrac{z+1}4, \quad y=3.$

  3. $-6x+3y+z=-9.$

  4. $3x-2y+z=10.$


Week 5

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!


Week 6

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) $\{(x,1-2x)\,|\,x\in{\mathbb R}\}.$

10) $\{(x,10-x)\,|\,x\in\mathbb R\}.$

11) The system has a solution (in fact, infinitely many solutions) only when $v=5u.$

Homework:
  1. #2 on p. 76.

  2. Take out your die and find all eigenvalues and eigenvectors for the four matrices listed on Day 8 (7 Feb). You should be able to do this geometrically with your knowledge of three-dimensional coordinates!

  3. Find all eigenvalues and eigenvectors of $\left[\begin{matrix}1&2\\5&4\end{matrix}\right].$
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.


Week 7

18 Mon, 5 Mar HW answers:

#3(c). $\lambda^2-1=0$

#8(c). Eigenvalue $\lambda=p$ with eigenvector $\left(\begin{matrix}1\\0\end{matrix}\right);$ eigenvalue $\lambda=q$ with eigenvector $\left(\begin{matrix}0\\1\end{matrix}\right).$

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!


Week 8

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

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 $A^{-1}:$ $7x+y=-2,$ $y-3x=8.$

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

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!


Week 9

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!


Week 10

26 Mon, 2 Apr Homework:

Solve the following system of equations by finding an LDU decomposition of the appropriate matrix. The system is

$\begin{align*}2x-y-z&=3\\x+y+z&=6\\4x-y+2z&=12.\end{align*}$

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. $a_n=3^n-5^n.$

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!


Week 11

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 five-card poker hands if there are no wild cards. Several explanations are given on the web page.


Week 12

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 odd-numbered 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 5-cent and 8-cent coins.
34 Fri, 20 Apr Finish the homework assignment on generating functions!


Week 13

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.