Review for the Second Midterm

Midterm II covers sections 6.1- 6.5, 6.7, 7.1. This is NOT a substitution for reviewing your notes, homework and quizzes that you took, but rather a complementary hand-out to help you organize the material that we covered.

Definitions:

- dot product in Rⁿ : two descriptions

- orthogonal basis

- orthonormal set

- orthonormal basis

- orthogonal projection

- orthogonal complement

- orthogonal matrix

- least-squares solution of a linear system

- inner product

- inner product space

- length, distance, orthogonality in an inner product space

- symmetric matrix

Concepts:

- Gram-Schmidt process

- QR factorization

- best approximation in inner product spaces

Theorems:

Ch. 6

- Theorem 2 (Pythagorean thereom) (p. 380) with proof

- Theorem 3 (p. 381)

- Theorem 4 (p. 384), with proof

- Thereom 5 (p. 385), with proof

- Theorem 6 (p. 390)

- Theorem 8 (Orthogonal Decomposition theorem) (p. 395) with proof

- Theorem 9 (The Best Approximation theorem) (p. 398)

- Theorem 10 (p. 399)

- Theorem 11 (The Gram-Schmidt process)

- Theorem 13 (Normal equations) with proof

- Theorem 16 (CS inequality) with proof

- Theorem 17 (Triangle inequality) with proof

Skills: you should be able to

- Compute orthogonal projection

- Find an orthogonal (or orthonormal) basis of a given vector space (i.e. perform Gram-Schmitd process)

- Compute QR factorization

- Find best approximation to a vector in a given vectors subspace

- Find least squares solutions, compute errors

- Diagonalize symmetric matrices

- Construct a (symmetric) matrix with given eigenvalues and eigenvector