Math 2B, Spring 2009 Home page

Green sheet

Class 21, Wed., June 17
We went over the Fibonacci numbers, and used matrices to find a closed formula for them.
We went over the four foundamental vector subspaces
Here are some nice problems on these subspaces.
We also went over section 3.8 on quadric surfaces
Homework:
Ch. 6.2: 1,2,3,5
Ch. 7.0: 1,2
Ch. 3.8: 1,2,3
Read section 8.1

Class 20, Mon., June 15
We went over material from the exam and the Gram-Schmidt orthogonalization process.

Class 19, Wed., June 10, 2009
We went over sections 6.0 and 6.1 and had exam 4.
Homework:
Ch. 5.3: 3,4,5
Ch. 6.0: 3
Ch. 6.1: 5,6,7

Class 18, Mon., June 8, 2009
We went over material on linear transformations from R^m to Rⁿ and also on change of basis matrices (see material mentioned for Class 17 in Jim Hefferon's text. Especially see the chapter on maps between spaces, especially the section on change of basis. Also see Chapter M of Beezer's text, especially the section on Column and Row spaces. Both texts treat these subjects more thoroughly than the one we are using.)

Class 17, Wed., June 3, 2009

Someone asked how to do problem #3 in 4.3. Here's an explanation.

Do the two homework problems in this handout.
In Hefferon's book, work these problems in chapter 2, section III:
3.17(b), 3.18(b), 3.19, 3.23, 3.24, 3,26.
There are several interesting applications of this material at the end of this section, on voting paradoxes, crystal structure, and dimensional analysis.

Turn in homework for chapter 4 on Monday.

We will go over the next two sections in the text.
We will also go over new material not presented thoroughly in the text.
Here is a free online text that I will use as a reference for this material,
Linear Algebra, by Jim Hefferon.
This link is to an ftp site, so you should be able to download immediately.
Especially read the sections on Vector Spaces and Linear Systems, section III.3.
The exercises in this text are linked to the solutions.

Another free online text is A First Course in Linear Algebra, by Robert Beezer
The section on Linear     Combinations (it's called section LC, the page numbers differ depending on which version you download) has very complete examples of what we'll talk about in today's class.

Class 16, Mon., June 1, 2009
We went over sections 4.4, 5.0 and 5.1.
Do all homework from those sections!
Ch. 4.4: 1-6, 9,11
Ch. 5.0: 1,2
Ch. 5.1: 1-4, 7, 8

Class 15, Wed., May 27, 2009
We went over sections 4.2 and 4.3
Homework:
Ch. 4.3: 1,2, 3,4,5

Class 14, Wed., May 20, 2009
Next exam is on Wednesday, May 27
We went over section 3.7, 4.0, and 4.1. Do homework
Ch. 3.7: 1a, 4,6,7,8,9,10a
Ch. 4.0: 1-5,9,12,13,17,18,19,23,26
Ch. 4.1: 1b, 2,3,4

Class 13, Mon.., May 18, 2009
We went over section 3.6
Ch. 3.6: 2-6,8,11,13-19,21,23,24

Class 12, Wed., May 13, 2009
We went over section 3.5.
Ch. 3.5: 1,5,6

Class 11, Mon., May 11, 2009
We went over section 3.4 and part of 3.5.
Do all HW problems in section 3.4

Class 10, Wed., May 6, 2009
We went over section 3.3.
Homework:
Ch. 3.3: 1-19
Additional homework: If you place a book with the spine along the z-axis, the front cover facing the positive x-axis, and the pages along the y-axis, then rotate 120 degrees around the line x=y=z, how will it end up? What if you do this rotation 3 times?

Class 9, Mon., May 4, 2009
We went over sections 3.1 and 3.2.
Homework:
Ch. 3.1: 1d,2d,3d,4d,5,6
Ch. 3.2: 1-4

Class 8, Wed., Apr. 29
We had exam 2, you did well on the scantron part!
Please do all of exercises 1-8 in section 3.0, mostly it's a review of 3D dot and cross product.
From now on we have to try to cover 2 sections per class meeting, so read ahead!

Class 7, Mon., Apr. 27
We went over concepts from the quiz, including eight symmetries in the
plane: reflections in the x- and y- axes, rotation around origin by Pi,
reflections in the line y = x and y = ˆx, rotations by +/- Pi/2, and the
identity. These make up a „symmetry group‰ or order 8, since the operation
of composition is closed, and it is possible to start with any one such
linear transformation, and given a second one of these 8, find a third
which will compose with the first to give the second (that‚s like saying
each one appears in each row and column of the multiplication table!) More
on this later.

We also went over section 2.7, in which we used the ideas of eigenvalue
and eigenvectors to
Easily raise a matrix with linearly independent eigenvectors to powers
Put a conic section (quadratic form) in standard form by rotating axes to
new axes that are in the direction of eigenvectors of the matrix
associated with the quadratic form.

Ch. 2.7 homework: 1,2,6,8-13
Turn in chapter 2 homework on Thursday!

Class 6, Wed., Apr. 22
We went over last part of section 2.5 and also 2.6.
HW:
Ch. 2.5, 1-19
Ch. 2.6, 3-10, one of 11/12/13, 15, 17
If you combine the first four operations we examined in Wednesday's quiz you get a "closed system," in which any operation followed by another gives one of those four. The four were: 180 rotation, reflection in y = x, reflection in y = – x, and the identity. If you add one more operation, such as reflection in the x-axis, then you must add three more, as you discovered in the second part of the quiz. Here's an additional homework problem: why, geometrically, do these operations make a "closed system?"
(a) Examine the total of 8 operations (linear transformations) geometrically and see if you can understand what is going on.
(b) If we start with  the function y = e^x, what is the result of applying each of these 8 linear transformations to that function?
(c) Make the multiplication table for these 8 operations, and find several patterns in the table.
Please write your answers to this homework problem on a separate sheet of paper, I will take this up as part of your homework!

Class 5, Mon., Apr. 20
We went over section 2.5. Do all homework problems in that section.
Class will have a new classroom beginning on Wednesday: S-35.
I hope it's big enough!
 
Class 4, Wed., Apr. 15, 2009
We went over more of section 2.3 and section 2.4.
New homework:
Ch. 2.3. 10-15,18
Ch. 2.4 1-12
Some errors: Exercise 7 on page 43 should read "... theta radians and phi radians..."
Exercise 14 on page 46 should read "Compare..." not "Conclude..."

Class 3, Mon., Apr. 13, 2009
We went over section 2.3 and some material on barycentric coordinates.
Exam 1 is Wed., Apr. 15.
There is a copy of the text on 1 hour reserve in the library, please make use of it.
Homework:
Ch. 2.3: 1-9

Class 2, Wed., Apr. 8, 2009
We went over sections 2.1 and 2.2.
Homework, by the way, is found throughout the text.
Ch. 2.1: 1,2,3,6
Ch. 2.2: 2,3,4,6,7

Class 1, Mon., Apr. 6, 2009
Went over material through 2.0.
Check out grading policy on green sheet...
Here is Jim Hefferon's free online linear algebra text.
It's a good secondary reference.
Homework due Wednesday:
Ch. 2.0: 1-5,7,1,,13,17,18,19,22