Review for the Final exam

Final exam is cumulative. It will cover sections from 5.1 -5.6, 6.1- 6.5, 6.7, 7.1-7.3. The list of concepts and theorems below is complementary to the list you got before the Midterm.

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:

- orthogonal basis

- orthonormal set

- orthonormal basis

- orthogonal projection

- orthogonal matrix

- least-squares solution of a linear system

- inner product

- inner product space

- length, distance, orthogonality in an inner product space

- symmetric matrix

- quadratic form

- positive definite, negative definite and indefinite quadratic form

- principal axes of a quadratic form

Concepts:

- Gram-Schmidt process

- QR factorization

- best approximation

- change of variable in a quadratic form

- constrained optimization problem

Theorems:

-         Theorems 19 – 34c from class.

From the book (note that most of them are duplicate to the ones we studied in class):

Ch. 6

- Theorem 8 (Orthogonal Decomposition theorem)

- Theorem 9 (The Best Approximation theorem)

- Theorem 10

- Theorem 11 (The Gram-Schmidt process)

- Theorem 12 (QR factorization)

- Theorem 13

- Theorem 16 (CS inequality)

- Theorem 17 (Triangle inequality)

Ch. 7

- Theorem 1\ - 6.

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 within a given vectors subspace

- Find least squares solutions, compute errors

- Diagonalize symmetric matrices

- Find principal axis of a quadratic form

- Make a change of variable in a quadratic form to get rid of the cross-product terms

- Classify a quadratic form (as positive definite, negative definite or neither)

- Solve a constrained optimization problem