Linear Algebra, 110.201, Spring 2004, W. Stephen Wilson

Problem Set Number 8, Due Thurs-Fri Mar 25-26.

These problems can be discussed in groups but must be handed in by each student. Keep in mind my warnings about how to learn in the Syllabus.

As discussed in class. If there is a weather closing for school that cancels a class then do the appropriate reading from the syllabus on your own and we will proceed in the next class as if we didn't miss anything.

This set is due at the beginning of section. There will be a quiz in section on this material.

The reading assignment is sections 5.4, and 5.5.

Problems:

Section 5.4:

10, 12, 16, 20, 22, 26

Section 5.5 is the most difficult and most important section in the course. Unfortunately, there are few problems of the sort I am interested in. (You will find many more on my old exams on the web.) Students tend to have a fairly good grasp of the dot product in Euclidean space and don't do that badly with understanding abstract linear spaces. However, the insertion of an abstract inner product on an abstract linear space seems to be more difficult to assimilate. Many students tend to try to force any old basis in a linear space to behave as if it were the standard basis in Euclidean space when it comes to the inner product. This is perhaps because such a basis does give an isomorphism of the linear space with the Euclidean space. However, this isomorphism does not usually preserve the inner product. In order for that to happen you must have an orthonormal basis in your linear space. To have that though you need to use the Gram-Schmidt process but, of course, to do that you had to understand the inner product correctly so we have come full circle. At any rate, the concept of orthogonal projection in an abstract linear space (with an inner product) is what allows us to do all kinds of important things. In particular, we see Fourier series as orthogonal projection (I will not emphasize Fourier series though). Likewise, approximating functions as polynomials is just orthogonal projection (this I will emphasize). Be forewarned that because I value this material highly my exams will be heavily weighted in this direction. My favorite problem below is problem 10.

Section 5.5:

4, 6, 8, 10, 14, 16, 20, 24