Sample Syllabus and Chapter Dependencies

The following outline represents what we cover in a standard, one semester course. One certainly could cover more. However, we feel that there is a general tendency to "overstuff" courses with material, resulting in less learning and less understanding.

If desired, determinants may be covered quickly, using formula (4) in Section 5.1 as the definition of the determinant. Then Theorem 1 and the theorems from Section 5.2 can be stated without proof.

There are no computer exercises for the determinant chapter. This omission is intentional for we want to stress that determinants are not a viable computational technique. Classes that incorporate a regular computer lab could use this time to do one of the application sections such as 1.5.1 or 3.4.1.

• Chapter 1 (10 classes)
  All sections. Section 1.4.1 is assigned as outside reading and Section l.5.l is assigned as a computer project.

• Chapter 2 (6 classes)
  All sections, except 2.2.1.

• Chapter 3 (8 classes)
  Sections 3.1 to 3.4. Also Section 3.4.1 may be assigned as outside reading.

• Chapter 4(6-8 classes)
  Sections 4.1, 4.2, 4.4 and (time permitting) one of 4.3 or 4 5.

• Chapter 5 (3 classes)
  Sections 5.1 and 5.2.

• Chapter 6 (6 classes)
  Sections 6.1,6.2 and 6.4.

• Chapters 1,2, and 3
  The sections in Chapters 1-3 are linearly ordered: each depends upon the previous. The only sections here which are not required for the rest of the text are the application sections (those with 3 digit numbers) and Section 3.5.
  If one wishes a "formal" discussion of function spaces, then either 2.2.1 or 4.3 should eventually be covered. However, there is enough "informal" discussion of function spaces in the exercise sets for other sections that students should be able to get a good feel for function spaces without covering either 2.2.1 or 4.3.

• Chapter 4
  All sections presume 4.1. Sections 4.2, 4.4 and 4.5 are independent of each other, while 4.3 requires 4.2.
  The material on least squares (Section 4.5) is (in our opinion) quite beautiful. One might consider covering the notions of dot product and orthogonality from 4.1 and then doing 4.5.
  The instructor should note that 4.1 is required for 6.2 while 4.2 and 4.4 are required for 6.4.

• Chapter 5
  This chapter depends only on Chapters 1-3 and is required only for Chapter 6. The material in 5.3 is optional.

• Chapter 6
  Chapter 6 requires 5.1 and 5.2. Additionally, 6.2 requires 4.1 and 6.4 requires 4.4.
  Section 6.3 is, with one exception, not required for the other sections. The one exception is that in the proof of thc Spectral Theorem in 6.4, we use one result (Proposition 1 in 6.4) which requires 6.3 for its proof. If 6.3 has not been covered, then Proposition 1 may, of course, be stated without proof.
  
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