The remaining sections from Lay that we will cover this quarter are listed below. In-class coverage will be light, as in my judgement, there is little that is completely new, especially to students who have been simultaneously studying physics. There may be problems assigned from these sections of Lay for homework, but even if not, when we encounter these ideas in QM, I expect that you will be able to use Lay as a resource.

• Section 5.5: Complex Eigenvalues. In particular, “Eigenvalues and Eigenvectors of a Real Matrix That Acts on Cⁿ.” Note that Example 6 and the Practice Problem at the end of the section reinforce question 8 from LA/QM Exam 1.
• Section 6.1: Inner Product, Length, and Orthogonality. Aside from terminology, little in this section should be brand new to you. A possible exception is the discussion of Orthogonal Complements, which nicely adds to our previous discussions of important subspaces, and sets up for sections 6.2 and 6.3.
• Section 6.2: Orthogonal Sets. This section introduces some new ideas though I think little that is counter-intuitive.
• Section 6.3: Orthogonal Projections. Theorem 8 and the discussion surrounding Figure 3 are the most important parts of this section. Please skip Theorem 9 and its surrounding discussion.
• Section 6.4: The Gram-Schmidt Process. The theory developed in sections 6.2 and 6.3 culminates in this section, which provides an algorithm for producing an orthogonal basis for any (nonzero) subspace of Rⁿ. If you need to be strategic with your time, this is the section to concentrate on, referring to 6.2 and 6.3 as needed to support your understanding of the Gram-Schmidt Process. Please skip the section on “QR Factorization of Matrices”.
• Section 6.7: Inner Product Spaces. This section generalizes the results from earlier. Please skip the section on Best Approximation in Inner Product Spaces. The section “Two Inequalities” discusses the Cauchy-Schwarz Inequality, which is called the Schwarz Inequality in Townsend QM Ch. 3. The section “An Inner Product for C[a,b]” extends the ideas of inner products to the important case of C[a,b], continuous functions on the interval [a, b], as defined by eq. (5) in Example 7. This generalization of inner product from the discrete case to the continuous case will be very important for us when we cover Townsend QM Ch. 6 at the beginning of spring.