Review for the FINAL EXAM II(Friday -12/16/05 1:00-3:00 PM).

CHAPTER 1,  Linear Equations and Matrices: Sections 1.2 - 1.5 
-Matrix operations: properties, powers of a matrix. 
-Special type of matrices: symmetric, skew symmetric, identity matrix, partitioned  matrices.  
-Nonsingular (invertible) matrices: properties.
-Linear systems and inverses.
-Review problems:
1.2 #  2,  8, 10, 11. 
1.3 # 10, 22, 30, 41, 43.
1.4 #  8, 10,  32, 36 
1.5 #  9, 12, 15, 18,  31, 32, 44.  
True or false (page 76) 1-9. 

CHAPTER 2, Solving  Linear Systems: Section 2.1 and  Section 2.2  
-(REF)  and Gauss Elimination Method 
-(RREF) and Gauss- Jordan Method
-Finding the inverse of a matrix 
-Characterize Nonsingular matrices (Result on page 104) 
Review problems:
2.1 #  6, 7, 14, 22,  33. 
2.2 # 8, 9, 10ab, 16, 18, 19, 20, 28, 30.
 True or false (page 122) 1-3 and 8-10.  Quiz(page 122): 1, 2, 3, 4. 

CHAPTER 3, Real  Vector Space. Sections: 3.2-3.5 and Section 3.8
-Vector Space: definition, properties
-Subspace: definition, span, null space of a matrix. 
-Span and Linear dependence/ independence.
-Basis and Dimension
-Vector spaces of dimension n: properties, know how to find bases. 
-Rank of a matrix
-Theorem 3.20 (page 210)
- Rank Theorem (page 211)
- Review problems:
3.3 #  4, 6, 8, 12bc, 14, 15, 28ab. 
3.4 # 2a, 4a, 8, 15b, 16, 20, 22, 24.
3. 5 # 2b, 4b, 8a, 12, 18, 20b, 24b, 33, 30, 34. 
3.8 # 13a, 17a, 26, 28, 29, 39. 
 True or false (page 217)  7-12 and 14-16.  Quiz (page 217): 3, 4, 5, 6, 7, 8, 10. 
 
 CHAPTER 4, INNER PRODUCT SPACES. Sections: 4.1, 4.3, 4.4, 4.5
-Length, inner product and magnitude in 2D and 3D.  
- Inner product spaces. 
  Definition, properties, distance, orthogonality, Cauchy-Schwarz  inequality 
-The Gram-Schmidt Process  
- Projection formula and projection theorems ( Theorems 4.9, 4.10, 4.11) 
-Review problems:
4.1 #  10, 25, 27 
4.3 #  10a, 16, 19, 30a, 32, 
4.4 # 2, 8, 10, 13, 23. 
4.5 # 2a, 13a, 20, 29
True or False 1-12, Quiz(page 286) 1, 2, 3, 6, 7, 8, 10. 
 
 CHAPTER 6  Determinants. Section 1-5
-Determinants, Cofactor  Expansion
-Properties of determinants 
-determinants and matrix operations  
-Inverse of a matrix using cofactors, Theorem on page 388.  
-Cramers's Rule 
-Review problems:
6.2 #  7c,8, 9, 14, 16, 17, 24, 26a, 30.  
6.3 # 3ab, 12; 6.4 # 2, 1;  6.5 #1, 3, 8. 
 True or False 1-7, Quiz (page 392) 1, 2, 4, 5, 7, 8. 

CHAPTER 7  EIGENVALUES and EIGENVECTORS: Section 1
-eigenvalues, eigenvectors, eigenspaces of a matrix
- Know to find the characteristic polynomial, the  characteristic equation and
 - Bases for eigenspaces of a  matrix.
 -When is a matrix diagonalizable? Know  how to Diagonalize a matrix.
 - Orthogonal matrices. Diagonalization of symmetric matrices.
-Review problems:
7.1 # 5ab, 7, 9bd, 17, 21, 22, 24ab.
7.2 #  6 bd, 8, 10ad, 12, 14a, 24
7.4 # 1,2a, 3, 4,  8, 9a, 15, 16, 19, 20
True or False (pages 492-493): 1-17. Quiz(Page 493) 5, 6, 7