Syllabus Math 253 F07

Linear Algebra with Applications (3rd Edition)

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ma253 —Linear Algebra Fall, 2007 Fernando Q. Gouvêa Linear Algebra is a crossroads, a place where a number of ideas come together. On the one hand, there is the circle of ideas concerning systems of linear equations and their solutions. On the other, there is Euclidean ge- ometry, especially the geometry of “straight” things like lines and planes. A third element is the theory of operators on function spaces, which arose out of the differential and integral equations of mathematical physics. Then there is the theory of matrices and determinants. Add to that some multidi- mensional coordinate geometry and a large dose of abstraction, and linear algebra is what comes out. The basic strategy in linear algebra is to start from the concrete objects (equations, vectors, functions, . . . ) and fit them all into a common frame- work: they are all elements of a linear space (sometimes also called a “vec- tor space”). Such spaces are (if you ignore the differences) all alike, and one studies them by trying to prove general theorems that will apply in all of the specific cases. This description highlights an important fact: linear algebra is a modern subject. Though many of the basic ideas are quite old, their unification into a single subject happened in the first half of the twentieth century. (Compare it with the calculus, which was in place, pretty much in the same terms as it is taught today in a typical calculus class, by the end of the 18 th century.) The basic approach of modern mathematics is abstraction , that is, we try to capture many different kinds of things with one big, general idea. Then we study that. There are many advantages to this approach. For one, you get many applications from one theorem. More subtle is “transfer of intuition:” once one realizes that, say, two-dimensional Euclidean space and spaces of func- tions share certain properties, one is able to use our good intuition about plane geometry to attack problems where our intuition is lacking. Abstrac- tion is also helpful because it “clears the underbrush” and allows us to focus Has anyone ever tasted an “end”? Are they really bitter?
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2 ma253 , Gouvêa, Fall 2007 on the more important features of a situation. And abstract ideas can have a very wide range of applications. The abstract nature of linear algebra is exactly what makes it so useful. But, of course, it also makes it difficult, at least at first glance. Helping you learn to think abstractly (and to understand thoughts expressed in abstract terms) is one of the major goals of this course. There is a mitigating factor, something which could not be predicted beforehand. Linear algebra turns out to be easier than most other parts of mathematics. This makes it even more useful: if we can formulate a problem in such a way that it becomes a problem in linear algebra, we have a much better chance of solving it.
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