Math 340 (Expectations)

One thing to keep in mind as you begin this course is that, while the calculus sequence is fairly computational, upper-level math courses courses focus more on theoretical and conceptual aspects of mathematics. While this course does include considerable practical and computational topics, it also deals with some very abstract ideas. You should not ignore these in favor of the computations.

For example, we will begin by defining a vector space in terms of several properties that it has. We will then use these properties to prove things like "0 + v = v for all v in a vector space". This will seem like a silly thing to do, as it is "obvious". Remember, however, that, as mathematicians, we are not so much concerned with determining what is true in an absolute sense, but rather what things follow from a given set of assumptions. The fact that 0 + v = v for all v follows from the definition of a vector space is an important conclusion about the consequences of defining a vector space as we did.

The process of determining the consequences of definitions like this is central to mathematics. It's the very heart of what we do, so you should pay particular attention to how this is done, and how each conclusion is reached. I recall very vividly a point in my own college career when I first began to realize the crucial nature of trying to prove a statement like the one above, and it was in a course like this one. As you work, look for these moments of insight. Don't let the computational aspects of the course become the main focus. In many ways, they are only there to illustrate the more fundamental concepts.

On the other hand, don't forget that the material is enormously practical. Most of the applications will not be presented in this course, but will show up in later ones (either in mathematics, or in another discipline), as we are just laying the foundations here. You should always keep in the back of you mind, however, that linear functions are used to approximate more complicated behavior, and that the mechanisms we develop for understanding linear objects, though abstract, are meant to be applied to any problem where a linear approximation is possible.

Math 340 (Winter 2008) web pages Created: 06 Jan 2008 Last modified: Jan 6, 2008 3:40:26 PM Comments to: dpvc@union.edu

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