Review List For Final Exam (December 17, 2007)

The Exam will cover the material covered in the Hour Exams and also Sections 6.1-6.6, 5.1-5.3, 5.8, 7.1, 7.4.

Definitions:

Inner Product, Length, Orthogonality

Orthogonal Complement

Orthogonal Set, Orthonormal Set

Orthogonal Projection

Orthogonal Matrix

Gram-Schmidt Processs

QR Factorization

Least-Squares Solution of Ax=b

Least-Squares Data Fitting

Theorem 5

Theorem 8 (The Orthogonal Decomposition Theorem)

Theorem 9 (The Best Approximation Theorem)

Theorem 12 (The QR Factorization)

Theorems 13 & 14

Theorem 15

6.2, 7, 11; 6.3, 1,3; 6.4, 9; 6.5, 1,5,15; 6.6, 3,9.

Sections 5.1-5.3, 5.8; Eigenvalues & Eigenvectors

Definitions:

Eigenvalue, Eigenvector

Characteristic Polynomial, Characteristic Equation

Similarity of Matrices

Diagonalizable Matrix

Power Method, Inverse Power Method

Theorem 4

Theorem 5 (The Diagonalization Theorem)

5.2,1; 5.3, 1,5; 5.8 , 1.

Sections 7.1, 7.4; The Singular Value Decomposition

Definitions:

Symmetric Matrix

Spectral Decomposition

Singular Values

Theorem 3 (The Spctral Theorem for Symmetric Matrices)

Theorem 10 (The Singular Value Decomposition)

7.1 , 13; 7.4 , 15.

Also, know how to solve Least-Squares problems using the SVD.