Math 415. Appled Linear Algebra
Midterm 2 Checklist

Chapter 3. Inner Products and Norms

3.1. Inner Products

• Know Definition 3.1 backwards and forwards and how to verify its properties in actual examples
• How is the norm of a vector defined in terms of the inner product?
• Be familiar with examples of inner products on Rⁿ and P⁽ⁿ⁾ and C⁰[a, b]

3.2. Inequalities

• Know the statement of the Cauchy-Schwarz inequality and how it allows us to define angles between vectors
• Given an inner product, know how to verify that two vectors are orthogonal?
• Know the statement of the triangle inequality and why it is true geometrically

3.4. Positive Definite Matrices

• Know the definition of positive definite as it applied both to symmetric matrices and quadratic forms
• The characterization in Theorem 3.21 is extremely important. You should know the calculations on page 153 that lead to it
• What conditions determine whether a 2 x 2 matrix is positive definite?
• What is meant by positive semi-definite? Know how to find null directions for a semi-definite matrix/form
• What is a Gram matrix? Be able to compute such matrices in simple examples
• Know the statement of Theorem 3.28

3.5. Completing the Square

• Know what is meant by the statement completing the square is the same as the LDL^T factorization of a symmetric matrix
• Review example 3.38

Chapter 4. Minimization and Least Squares Approximation

4.1. Minimization Problems

• Be able to describe the closest point problem geometrically and write it down analytically as a minimization problem

4.2. Minimization of Quadratic Functions

• Be familiar with the general quadratic function on page 186 (equation (4.10)) and Theorems 4.1 and 4.4 that tell us how to minimize it
• Review Examples 4.2 and 4.3 as prototype examples of how to do the minimization in practice

4.3. Least Squares and the Closest Point

• Know how to solve the closest point problem in concrete examples
• What is the least squares solution of a linear system? What are the normal equations for finding the least squares solution and how do they arise?
• Review Example 4.9

Chapter 5. Orthogonality

5.1. Orthogonal Bases

• Know what it means for a collection of vectors to be mutually orthogonal and how this relates to linear independence
• What do we mean by an orthogonal/orthonormal basis? How do you turn a vector into a unit vector?
• Be able to find the coordinates of a vector relative to an orthogonal/orthonormal basis

5.2. The Gram-Schmidt Process

• Know what the Gram-Schmidt process uses and produces, and how to execute it in simple examples (including one involving polynomials)

5.3. Orthogonal Matrices

• Know the definition and basic properties of orthogonal matrices (products, determinant, connection to orthonormal bases)
• Know the form of rotation matrices about standard axes in 2 dimensions

5.5. Orthogonal Projections and Least Squares

• Know the orthogonal projection formulae (5.63) and (5.64) and when and how to use them
• Know Theorem 5.39 and be ready to apply it in simple cases

5.6. Orthogonal Subspaces

• Know the definition of orthogonality of two subspaces and orthogonal complement
• Be able to find orthogonal complements (that is, bases of them) in simple cases
• Theorem 5.55 is vital since it tells you how to deduce whether the vector b in Ax = b admits solutions, so be able to find bases of cokernels!!
• Know Theorem 5.59 and Equation (5.82) for finding the unique solution of minimum norm of a system with multiple solutions