Week 5: March 8-12

Chapter 5.1 Orthonormal bases and orthogonal projections

Key points: a) Recall definition of the dot product, orthonormality, orthogonal

complement.

b) Orthogonal projections

c) Triangle inequality: |x + y| ≤ |x| + |y|.

d) Pythagorean theorem

e) Fact 5.1.7: This is the whole point!

Homework: 6, 26, 28, 36, 38, 14*, 29*

Chapter 5.2 Gram-Schmidt and QR factorization

Key points: a) Don’t stress QR factorization-spend the time on Graham-Schmidt

and why it works.

Homework: 4, 14, 32, 34, 40

Chapter 5.3 Orthogonal transformations

Key points: a) Orthogonal transformations preserve lengths so all dot products

b) Columns of an orthogonal matrix are orthonormal (and why)

c) The tranpose of a square orthogonal matrix is its inverse

d) Products of orthogonal matrices are orthogonal

e) Relations between transpose and inverse

f) Symmetric and skew-symmetric

g) Matrix for an orthogonal projection.

Homework: 8, 10, 14, 18, 20, 40*, 44*