Math 311-200 — Final Exam Review

Extra office hours   I will have extra office hours on Friday, from 9 am to 10:30 am, and on Tuesday, from 9 am to 10 am and from 2 pm to 3:30 pm.

Material not directly covered

• There will be no direct questions on matrix operations, row reduction, determinants, etc. However, you will need to know these topics well enough to use them in answering questions from the material below.

Material directly covered

• Chapter 2, sections 2.1-2.4
• Chapter 3, sections 3.1, 3.2, 3.3, 3.5 (skip Theorem 3.6)
• Chapter 4, sections 4.1, 4.2, 4.3, 4.5, and 4.6. You should also know the structure of the characteristic polynomial; that is, that
  p_A(λ) = λⁿ - (trace(A))λ^n-1 + ... + (-1)ⁿ det(A)
  
• Chapter 5, section 5.3
• Chapter 6
  • Inner products and inner product spaces
  • Orthogonal and orthonormal sets
  • Gram-Schmidt process
  • Least-squares
  • Fourier series
  • Discrete Fourier transform

Calculators   Calculators will be allowed.

Structure   There will be 7 to 9 questions, some with multiple parts. Material from chapters 2-5 will count for 65-70 points, and material from and related to chapter 6 will count for 30-35 points. The questions will be similar to ones done for homework or as examples done in class or in the text. There will be one ``theory'' question taken from the following list.

1. Be able to prove this representation theorem: If a vector space V has a basis B = {v₁, ..., v_n}, then every v in V can be represented in one and only one way as v = c₁ v₁ + ... + c_n v_n, where the c_j's are scalars.
2. Be able to show that if a vector space V has a basis B = {v₁, ..., v_n}, then every set with n+1 or more vectors is linearly dependent.
3. Be able to state and prove Schwarz's inequality.
4. Be able to show that if a subspace of an inner product space has an orthogonal basis B = {v₁, ..., v_n}, then for any v in that subspace the coefficients in the expansion v = c₁ v₁ + ... + c_n v_n are given by
   c_j = < v_j | v >/< v_j | v_j >.

Updated 5/2/06