What is linear algebra? Functions and equations that arise in the `real world' often involve many tens or hundreds or thousands of variables, and one can only deal with such things by being much more organized than one typically is when treating equations and functions of a single variable. Linear algebra is essentially a `language for accounting' that's been developed just for this purpose. We will learn methods for solving equations and ways of understanding their solutions that are very effective when the equations are what is called (of course) `linear'. In a kind of analogical way, we will even learn to `visualize' many-dimensional situations.
What are differential equations? Many functions that come up in applications do so only in an indirect fashion. That is, rather than being told what the formula is for a function, one is given some (differential) equation relating the function to one or more of it's derivatives. For instance, a bank does not advertise a formula for the amount of money in a hypothetical account. Instead it advertises an interest rate, which is a way of saying how the amount of money in an account will change with time. The main goal in studying a differential equation is to understand the function it applies to. In simple situations one can use the equation to determine a formula for the function. In more complicated ones, when formulas are impractical or impossible, one can still try to answer specific question, like `what happens to the function when the independent variable becomes large? does the function also become large? small? ' etc.
What we'll cover: we'll
6 in Lay's book.
The remaining 1/3 of the
semester (and the entirety of math 325, should you take it) will be
spent on differential equations. In this semester, we'll cover
chapters 1 through 3 of Boyce and DiPrima. Time is short, and we won't
have much time to discuss applications of the math we're learning,
particularly the linear algebra. So I would highly encourage you
to look at the chapter intros and some of the 'application' sections
(e.g. 1.6) in Lay's book. One can plausibly argue that linear
algebra is *the* fundamental tool in modern applications of
mathematics, used to determine airline schedules, rank webpages in
search engines, compress electronic data, model the flow of oil
underground, and so on and on and on.
How you will be evaluated:
5% of your final grade, collected each Friday except
during exam weeks. If I get ambitious, I'll also give two or
assignments separate from textbook homework. Should that happen,
I might weight homework higher (e.g. 10%) in your final grade,
borrowing from the weight I assign to the final.