I don't think it is possible for me to cover a whole section of this linear algebra book in one class. Today, we started the section on Elementary Matrices and just a little ways in, I knew it was going to be a difficult section.
The main purpose of this section is to use the concept of matrix multiplication to perform row operations and demonstrate the basic results of row equivalence, such as categorizations of nonsingularity: (i) that nonsingular matrices are row equivalent to I, (ii) that the system has a unique solution. Also, elementary matrices provide a way to calculate matrix inverses through row reduction.
However, a good many of the results require a level of proof that is new to the students. I know that many of them are planning on going into engineering, another group is planning on teaching either middle school or secondary mathematics, and last of all, a group will likely go to graduate school in mathematics. It's tough to design a course in linear algebra to meet all their needs but in the end, it's worthwhile to see the reason behind the method.
Today in class, the rest of the students finished presenting the short oral reports over the classical arguments for the existence of God. (see the previous post). Following this, we delved into the basics of Logic and Reason, covering the basic structures of logic: propositions, conjunctions, disjunctions, negations, conditional and equivalence connectives. Following a presentation of basic rules of logic, I got a little ways into the Rule of Inference, covering Modus Ponens, Modus Tollens, Hypothetical Syllogism, and Disjunctive Syllogism. There are a few more and then we will cover fallacies, divided into three main categories:
- Fallacies of Relevance
- Fallacies of Presumption
- Fallacies of Ambiguity
A colleague of mine will join the class for a few days to help us cover an overview of the History of Science and how science develops. We'll consider in detail the scientific method. After which we are ready to start with the heavy issues of Origins.