# Linear Algebra: Inverse and Transposes

It took three classes but I finally finished all of the section over matrix algebra. During today’s lecture we walked through the concepts of matrix inverse and matrix transposes. It’s amazing how long these lectures stretch out when you choose to demonstrate the concepts with examples involving matrix operations. As far as I can tell, all the students are following along very closely, except when I start going off on a tangent and trying to draw connections between linear algebra and higher level mathematics.

Somehow I did get sidetracked today into talking about my days as an undergrad at Wayland. I recounted the story of my first day of class with Dr. Almes, when he called out several names at the beginning of class, stating that he wanted to see all of us after class. I was actually filled with dread. I just knew that I had already done something to upset my professor. He was practically the head of the math department and I had ruined my chances of making nice with him. What was he going to do to me? I was going to have to change majors again.

My fears were relieved when, after class, he simply thanked us for attending his church the previous Sunday and invited us to return. It hit home the fact for me that Wayland was more than a typical school. It was a family. It was a Christian family where I was going to school with my brother’s and sister’s in Christ and I was being taught by my brother’s and sister’s in Christ. There was no need to separate my learning from my faith, and in fact, in many of my classes they would be closely integrated.

Next time in Linear Algebra, we will cover elementary matrices and derive a computational method for finding matrix inverses.

# Calculus IV: Exam I

During Calculus IV on Tuesday, we had Exam I. I’ve decided that I’m not generally very good at producing tests at the appropriate level for the course. Usually they are much too hard (1 or 2 problems that require a LOT of tedious calculation) or too long (too many problems). The reason is not that I am particularly mean-spirited but that the my writing process is flawed. I start by writing a review for the test. I generate a solution guide for the review. Then, I come back and make a few minor modifications to the review, as well as adding a couple of new extensions of the problems on the review. The end result is that those minor modifications generally result in something not simplifying where it did before the mod.

With all that said, I was pleased with the Exam I that was given in Calculus IV on Tuesday. It tested over the set of material I was most concerned about. It was just the right length. All the students finished completely and handed in the test within the last 5 – 10 minutes of class. A couple problems were sufficiently difficult to challenge them but they walked away feeling like they had done fairly well. Of course, I have not graded them yet and may have a completely different perspective after that. But for the sake of confidentiality, you will never know. Sorry. But you can look at the test here: calculus-iv-exam-i.pdf

We will begin looking at Partial Derivatives in the next class. Yeehoo!

# Intermediate Analysis: Cardinality

During I.A. on Tuesday (Holy Cow 7:25 am is early to start a class) two students presented problems at the board. I really enjoy having this part of the class. All of these students have taken courses from a colleague of mine. She has made oral presentation of homework a regular part of her upper level courses. So, they are all used to it, but it is fairly new to me. I enjoy hearing the thought processes of the students as they assimilate this knowledge. I even developed Rubric to grade these presentations. Don’t I sound like an education geek? (By the way, I was never formally trained to teach.)

We then covered a few more results regarding countability, such as categorizations of countability, the fact that the countable union of countable sets is countable, and the countability of the rationals. We then diverted from the text a bit and introduced the concept of cardinality and cardinal numbers. I introduced them to aleph-null as the “size” of the natural numbers and the smallest of all transfinite cardinal numbers. We, then, proved that the “size” of the reals is strictly larger than the set of natural numbers, i.e., the Reals are uncountable. I ran out of time before covering Cantor’s Theorem. It is one of my favorite proofs in this class so there’ll be an entry in the next day or two over just that. I’d describe it as “slick.”

Next time, we finally finish Chapter 1 and begin covering The Real Numbers. What the heck are they anyway? We’ll using the axiomatic approach.

# Linear Algebra: Properties of Matrix Algebra

During Linear Algebra on Monday, I began class by answering homework questions. I clarified the fact that $(0, \alpha, -\alpha)$ and $(0, -\alpha, \alpha)$ represent the same general solution to a linear system when $\alpha \in \mathbb{R}$. Somehow the discussion of applications of the techniques we are learning came up. Thus far, we have basically covered how to use Gauss-Jordan elimination to solve linear systems. I pointed to examples from engineering (such as structural analysis of trusses) and computational fluid dynamics (discretization procedures to solve PDEs). I understand that these are a little outside the scope of this class but I chose them because of the individuals asking the question and based on their particular interests in mechanical engineering and aeronautics.

After chasing a couple rabbits, we continued working with operations on matrices, namely matrix addition, scalar multiplication and matrix multiplication. We went over the properties of the operations such at commutativity of addition, associativity of all three, distributive laws, etc. We proved a couple of these.

I recognize that for many of these students, they have little or no background in formulating a formal proof. In light of this, I chose to first prove that matrix addition commutes. Then I chose to show them one of the longer (thought not really much more difficult) proofs, the associativity of matrix multiplication. The real challenge was to help them see through the cumbersome notation, that it simply hinged on the associativity of multiplication of real numbers. I’m not sure that next time I teach this class I would want to bother with this proof. I think it is important that they see proofs of these fundamental concepts but I can accomplish that with a couple of simpler ones.

We just got into inverses and identities and will finish up this section on Matrix Algebra next time. We’ll then be ready to start talking about partitioning matrices and the doing some applications, such as traffic flow, balancing chemical equations, and search engines.

# Faith and Science: Finishing Definitions

During Faith and Science on Monday, the students were allowed to continue collaborating in their groups (2 groups of 4) to decide on their final working definitions of faith and science. After 30 minutes or so, the groups went to the board and recorded their definitions for faith and for science. They then made the effort to assimilate their definitions. We all recognized that within different contexts, different meanings of the terms are appropriate. The definitions below reflect what we will use as our default definitions of these terms as we debate the interaction between the two. If we choose to refer to variations on these definitions, we will be required to explicitly say so. Otherwise, this is what we mean:

Faith is . . .
The belief in the supernatural or God intervening or being involved in the lives of humans and our world (in ways science has been able to explain AND those that cannot be fully explained by science).

Christian Faith is . . .
Absolute belief, trust and loyalty to God and His promises through Christ to salvation.

I should note that one group wanted a more general definition of faith while the other thought it more appropriate to use a definition that reflects the way THIS class will be using the term. Our perspective is primarily how the Christian faith interacts with science. We decided to use both definitions, added the term “Christian” to the second group’s defined term.

Science is . . .
a systematic method for pursuing and acquiring knowledge of aspects of our universe, formulated through human reason.

Beginning next class we will begin studying logic and reason, laying down the specific rules for logic, and analyzing arguments.

# Faith and Science: Definitions

To begin class on Friday, I went over a few interesting entries that have been made on the class blogs. As part of this course, the students are required to maintain a personal blog over the topics of this course. Up to this point, they have each been asked to post entries on a series of questions, such as encountering conflicts between science and religion, their perspective on miracles, their firm or nor-so-firm opinion on the origin of the universe, etc. In the past (this course having been taught twice before), we have simply had the students fill out a questionnaire handout. The blogs allow the students to read one another’s entries and make comments. I have also been making comments and have found their participation, thus far, to be very encouraging. Fortunately, it is a fairly small class so I can manage to read each of their entries in detail.

Following this review of some of the interesting postings, we entered into our discussion on the definitions of the terms we will be using throughout the course. In order to be very precise in our debates we want to establish specifically what we mean by the terms Faith and Science, as well as some related terms such as Religion, Natural, Evolution, Creation, etc. In many cases, a debate can be made fruitless when two sides use the same term to mean completely different things. For example, we can propose a definition for Science that basically makes it the study, through observation and experimentation, of the natural world. That works just fine but you must specify what is meant by the natural world. One might define natural in such a way to exclude any influence from the supernatural. If so, one decides from the very beginning that any “scientist” that studies how God might be revealed in nature or how the natural world points to a supernatural origin cannot be doing science. We have yet to discuss those issues so we’ve made not conclusions about the validity of such theories as Intelligent Design or even Theistic Evolution. And yet, how we define science can exclude it from the very start if we are not careful.

I had the students prepare for class today by searching for various definitions of the above terms and they were asked to identify the important components of each definition that they feel are vital to a precise and foundational definition of the term. We focused primarily on Faith, Science and Natural. After collecting many varying components, writing them on the board, and discussing, the class was divide into two groups and asked to formulate their working definitions. Class ended in the middle of this activity and we will pick up there next time.

# Pecking order of the Sciences

One of my students in Linear Algebra reminded me of quote I had heard some time back concerning the pecking order of the sciences. The version he quoted was what he had heard from a colleague of mine here at Wayland:

All chemists want to be physicists.
All physicists want to be mathematicians.

Although, I couldn’t recall it at the time. I looked it up and found the two forms of this quote I have heard before:

or

The biologist wants to be a chemist.
The chemist wants to be a physicist.
The physicist wants to be God.
God wants to be a mathematician.

No, I do not have a God complex. But who could blame me. I am a mathematician.

# Faith and Science: Up to this point

So far in the Faith and Science course we have spent some time discussing the importance of our worldview, Flatland, and definitions of important terms. We concluded that even scientists who hope to achieve true objectivity are inevitably victims of their own biases. Are there are ways to overcome this bias? Certainly there are areas of scientific endeavor where the worldview affects ones understanding or interpretation of a particular result greater than others. In the end, worldview matters. Scientists should be aware of their bias and not necessarily try to hide it behind an attitude of objectivity. In particular, I can say that as a Christian I am motivated to understand God’s creation by my desire to learn more about the Creator. And yet, the truth of the topics covered in, say, my Intermediate Analysis course are true and independent of my personal beliefs. Our expectation of how the world “works” has its source in our own beliefs about the world around us. This can affect, and most likely does, the way we “do” science.

God has given the gift of a rational mind to humanity so that we can use it to arrive at the truth about this world. I’ll not recreate the entire discussion here but it was very intriguing.

Next, the class read a short little book written in the late nineteenth century, by Edwin A. Abbot (although originally under the pen-name of A.Square), called Flatland. It tells the story of the life a square who lives in a two dimensional world called Flatland. The first third of the book describes the the life of a figure in two dimensions. Imagine taking a penny and placing it on the table and then lowering your eyes to the level of the top of the table. You see that the circle that was the penny has become a straight line. From that perspective, all objects in the plane have become straight lines. It is very interesting world that Abbot creates. The rest of the book discusses the adventures of A.Square into the worlds of Lineland and Spaceland. He visits Lineland, a one dimensional world and tries, in vain, to explain the nature of his two dimensional world. Later, a sphere visits him from Spaceland, a three dimensional world like ours. A.Square cannot not understand what the third dimension is like until he visits the world and sees Flatland from above. When he returns to Flatland and tries to explain the nature of the third dimension, “upward, not northward”, he is outcast and thrown into prison for his heretical ideas.

We used it as an important lesson on how we communicate with individuals with an entirely different set of assumptions about the world around us. Very often, we not only don’t understand their language but their lack of similarity to our experiences make our ideas seem completely foreign and incomprehensible. We must make every effort to understand each others arguments accurately as we debate. And of course, we critique ideas and not persons. In fact, on the first day of class we laid down the parameters of our debates in our class and that principle was high up on the list.

# The Amazing Feats of Hot Glue

What the heck is it? I considered making my own contribution similar to Little David’s series of the “Watzit” contest. I decided that since this is likely to be a one time entry that you could just pretend to not know what it is and pretend not to see the rest of the pictures below.

Thanks to Lifehacker, I have answered an age-old question: How do you replace those lost rubber feet off the bottom of your laptop? Every laptop, I’ve ever owned has had little rubber feet on the bottom that basically act as “scootch” guards, keeping them from sliding around, as well as elevating the laptop to help keep the temperature down slightly. Every laptop I’ve ever owned has also chosen to reject those feet at some time and, inevitably, they are lost for all time. Once or twice, I have sent a laptop in for repairs for other issues such as a broken screen or crashed hard drive and occasionally, the nice repair folks at Compaq, HP or Dell would replace those for me. Not always, but at least a couple of times. I’ve also trip purchasing the little rubber feet that stick on with an adhesive and although they work for a time, they eventually are rejected as well and lost again.

I’ve discovered a new idea. Someone mentioned using hot glue to adhere these cheap little pieces of rubber so as to keep them on for longer periods of time. As a matter of fact, I’ve tried all the glues I felt comfortable sticking to the bottom of this expensive piece of machinery and they all fail in a relatively short amount of time. Then someone realized, who needs to buy the rubber feet when the glue will work just as well. And if it does happen to fall off, just put another dab on. See my handiwork:

Other amazing feats of Hot Glue (pun intended):

• Weak bonding especially good for porous medium
• Shoe repair
• Handitack replacement, such as attaching laminated posters to cinder block walls. You should probably not use on painted drywall unless you’d like to remove part of the wall when the poster comes down. (but who would ever take down that Princess Poster?)
• Permanent Ear Plugs
• Keeping the kids down for a nap
• Any other ideas???

# Calculus IV: Functions of Several Variables

Having just finished a very short chapter on vector functions, we began the chapter that will cover Partial Derivatives. Before we started that lecture, I took a little class time to answer a homework question. It was one of my favorites. Having covered the concept of curvature, the students were asked to find a polynomial, $P(x)$ such that the function

$$f(x) = \left\{ {\begin{array}{*{20}c} 0 \hfill & {,\,x \le 0} \hfill \\ {P(x)} \hfill & {,\,0 < x < 1} \hfill \\ 1 \hfill & {,\,x \ge 1} \hfill \\ \end{array}} \right.$$

is continuous, smooth and has continuous curvature. Not hard but still fun to do. Feel free to try it yourself and post the answer in the comments.

We, then, began functions of several variables talking about four ways to describe a function of two variables and then looking at examples of each. (Verbally, Numerically, Graphically, Algebraically) We plotted a few interesting surfaces in Maple 10 and then began discussing level curves (contour maps). Using examples such as weather maps (temperature, pressure, etc.) and topographical maps helped the students get a good feel for how we can look at a 3 dimensional graph in a two dimensional plane (without necessarily resorting to perspective drawings). We’ll start looking at level surfaces, or iso-surfaces, to generalize to functions of three variables. For example, how might you visualize a 4D hypersphere by using iso-surfaces?

For future versions of this lecture, I could definitely improve the lecture by bringing resources to class like video clips of a weather forecast or actual topographical maps of this area. Also, if I have time I’d like to have a demo of the software Viz5d so that I can visualize some data sets with animated iso-surfaces. I used to run a demo of a weather simulation when I worked for the High Performance Computer Center of Texas Tech University. I animated iso-surfaces of water vapor levels in the atmosphere. The result was a three-dimensional model resembling cloud movement.