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