**HMC Math 63: Linear Algebra II (Summer, 2002)**

Homework

**
FINAL: Due Saturday, June 29
**

### Assignment #13 (due Fri. 6/28):

On Wednesday, we discussed using Iterated Function Systems (IFSs) to draw fractals like
Sierpinski's Triangle and
Koch's Snowflake.
Now you're going to write a MATLAB function to draw these fractals from the IFS functions. Here are some files you'll need.

**13A:** Write a MATLAB m-file to define a new function `frac(C,k)` that takes the matrix/vector data of an IFS in a matrix C and generates the image (with k points plotted) of the corresponding fractal.
frac.m contains an outline and some code, as well as lots of notes and hints.

**13B:** Test your `frac(C,k)` function on the data variables `Sier` and `Koch`. You can load these variables into the workspace by executing the m-file hw13.m.

**13C:** Now try your `frac(C,k)` function on the data variables `Frac1`, `Frac2` and `Frac3`, also in
hw13.m.

**13D:** Here are drawings of two more fractals...
Frac4 and
Frac5. Try to determine the IFS that generates them. That is, find a 2n-by-3 IFS matrix C for each so that `frac(C,k)` generates the fractal shown. Use your `frac(C,k)` function to test your guess.

**13E:** Now make your own... draw some squares and/or parallelagrams in the unit square that you think might make an interesting fractal. Compute the IFS data matrix C for it, then have `frac(C,k)` draw it. I don't care if it's pretty... as long as it's original. (I do want at least 3 functions in your IFS; that is, C should be at least 6x3.)

**Fun:** Here's some more IFS data I've made...
fun.m. If you're enjoying playing with your new toy, these are a few IFS data sets that generate interesting results. However, you needn't submit them with your homework... you will receive no credit for them, and we already know what they look like.

### Assignment #12 (due Wed. 6/26):

### Assignment #11 (due Tue. 6/25):

**Section 5.6:** 2, 8, 17ab

**Section 7.1:** 13, 19, 23, 26, 32

**9A:** Okay, this was too hard for me, but I'll bet you can do it....

Let (I-A)' mean "(I-A) inverse". Assuming that this exists, show that x=(I-A)'b is a fixed point of the multidimensional discrete dynamical system x(k+1) = Ax(k) + b.

**9B:** We can transorm the system x(k+1) = Ax(k) + b into the simpler system u(k+1) = Au(k) with a simple change of variables. Show that the change of variables u(k) = x(k) + (A-I)'b works so long as (A-I) is invertible. That is, given x(k+1) = Ax(k) + b, show that u(k+1) = Au(k) also then holds.

**
QUIZ II: Due Monday, June 24
**

### Assignment #10 (due Mon. 6/24):

**Section 5.4:** 1, 3, 5, 9, 11, 17, 28, 31

### Assignment #9 (due Fri. 6/21):

**Section 4.2:** 31, 33, 36

**Section 4.4:** 21, 22, 23, 24, 28

**Section 4.5:** 31

**9A:** Prove the following result from class:

For a linear transformation L:V->W where dim V = dim W, L is 1-1 iff L is onto.

### Assignment #8 (due Thu. 6/20):

**Section 1.7:** 17, 21, 26, 31, 36[M]

**Section 1.8:** 3, 11, 16, 25, 27

**Section 4.2:** 34, 35

### Assignment #7 (due Wed. 6/19):

**Section 4.4:** 13, 19, 25, 35

**Section 4.7:** 1, 3, 9, 13

**Section 6.2:** 29

**7A:** Let [**u**]_s denote the coordiante vector of **u** with respect to basis S.

Given vectors **u,v** and real scalar c, prove that

*(i)* [**u+v**]_s = [**u**]_s + [**v**]_s
*(ii)* [c**u**]_s = c[**u**]_s

### Assignment #6 (due Tue. 6/18):

**Section 6.1:** 19

**Section 6.2:** 12, 15

**Section 6.3:** 1, 8, 19

**Section 6.4:** 10

**6A:**
Write an m-file to define a new MATLAB function "gs(A)" that applies the Gram-Schmidt process to the columns of a matrix A. All the following work should be done in MATLAB.

Further instructions (necessary)

Excessive hints (optional)

gs.m (rough m-file template... recommended)

| 1 2 3 0 |
A = | 3 0 -3 2 |
| -2 1 4 -1 |

**(i)** Apply your gs function to A. Let B=gs(A).

**(ii)** Verify that B's columns are orthonormal by computing B'*B. What should the result be, and why?

**(iii)** Verify that B's columns span the column space of A by solving Bx=v for each column v of A (use rref([B A(:,j)]) for each column j). Briefly explain the zero values in each such solution.

**(iv)** Use MATLAB, and specifically your gs function, to do Problem 6.4.22.
When you are finished, e-mail your gs.m and diary file to the grader at
Shane_Markstrum@hmc.edu with subject line
"Math 63 HW#6 - (your name)" (where, of course, "(your name)" should be replaced with your actual name). You'll turn in the written portion of your homework Tuesday in class as usual.

**
QUIZ I: Due Monday, June 17
**

### Assignment #5 (due Mon. 6/17):

**Section 6.1:** 6, 23, 29, 30

**Section 6.2:** 10

**Section 6.4:** 2

### Assignment #4 (due Fri. 6/14):

**Section 5.3:** 6, 10, 11

**4F:** Prove that the similarity relation is *transitive*,
that is, if A ~ B and B ~ C then A ~ C.

In this assignment, you will learn to use MATLAB. All instructions for this assignment are contained in

hw4.gif.

Supplimentary material can be found in

hw4.m.

diary.txt.

Note that you may work together on the MATLAB parts of this assignment.
Please read the instructions in hw4.gif before beginning the assignment.

### Assignment #3 (due Thu. 6/13):

**Section 4.3:** 21, 25

**Section 5.1:** 6, 15, 25, 29

**Section 5.2:** 6, 14, 21

**3A:** For the 4x5 matrix A from Problem 4.3.14 (pg 238),
find bases for the null space, row space and column space of A.
For each x in your basis for rs(A) and each y in your basis for ns(A), verify that x and y
are orthogonal (perpendicular).

### Assignment #2 (due Wed. 6/12):

**Section 3.1:** 4, 9

**Section 3.2:** 11, 25

**Section 4.1:** 5, 7, 8, 32

**Section 4.3:** 11

### Assignment #1 (due Tue. 6/11):

**Section 1.5:** 6, 14 (read both before doing either), 21

**Section 1.6:** 28, 30, 34, 35

**Section 2.1:** 22 (Hint: Consider ABx)

**Section 2.2:** 4, 24

**Return to: Math 63 Main Page
* Greg Levin's Page
* Department of Mathematics**

Email: *levin@hmc.edu*

On Deck...

### Assignment #13 (due Never):

This is not a real assignment... just a few computational

problems that you could do to practice for the final.

NOTE: Some of these have appeared on previous assignments.

**Section 4.2:** 31

**Section 4.4:** 13

**Section 4.7:** 5

**Section 5.4:** 5, 11

**Section 5.5:** 5

**Section 7.1:** 17