Review for the FINAL EXAM  (Thursday-December 16, 10:30-12:30).
I) Review the the Midterm I and Midterm II. 

II) From Leon's book: 
CHAPTER 3- VECTOR SPACES (Section 1-Section 4)
-Definitions, Properties, Vector spaces axioms 
-Subspaces, Span and Spanning sets.
-Linear Independence (Dependence)
-Basis and dimension 
-Review problems:
3.2 # 2, 5, 11, 14
3.3 # 2, 6b, 14                      
3.4 #3, 4, 8
Chapter Test (page 186) 5, 6, 7 

CHAPTER 4- LINEAR TRANSFORMATIONS (Section1, Section2)
-Linear transformations. Definition and properties.
-Image and Kernel, Matrix representation
-Review problems:
4.1 #4, 8, 16c, 24
4.2 #2c,  8, 16b
Chapter Test (page 221) 2, 3, 4

CHAPTER 5- ORTHOGONALITY  

5.1 Scalar product in R^n, norm, properties. 
 Scalar and vector projections.
 Review problems: 1, 2, 3, 16

5.2 Orthogonal subspaces
-Orthogonal complement.
-Applications to matrix operators.
 Relate N(A), R(A), N(A^T), R(A^T).
Review problems: #2a , 6, 7, 8

5.4  Inner Product spaces:
 Inner product, norm-properties. 
 Cauchy-Schwarz inequality
 The angle between two vectors.
 Review problems: #2, 13, 24

5.5 Orthonormal sets.
 Orthonormal , orthogonal sets of vectors 
 Parceval's formula
 Orthogonal Matrices (Definition, properties)
 Review problems: # 4, 6, 12, 15, 25, 31

5.6 Gram Schmidt Process.
Review problems: #3, 4, 11

 Chapter Test (page 311) 4, 6, 8, 9

III) From Edwards-Penney book

CHAPTER 3 Power Series Methods -  
3.1 Power Series  
-Radius of convergence, the method
 Review problems:    # 6, 20, 22

3.2   Solutions near ordinary points
  Find the minimal radius of convergence using Theorem I (page 207)
 - Initial value problems.
  Review problems:  # 8, 24, 28

3.3 Regular singular points
-definitions,  the method of Frobenius
-Review problems: # 2, 4, 17

3.4  Bessel's Equations
-The Gamma Function
3.6 Applications of Bessel Functions (theorem I)
Review problems: #6, 7, 8

CHAPTER 4   Laplace Transform Method Sections : 4.1-4.5 

- Laplace Transform and the Inverse Laplace Transform.  
-Definitions, properties. The table of Laplace transforms. 
-Transformations of initial value problems
-Partial fractions, convolution of two functions.
- Periodic and piecewise continuous input functions
Review problems: 
4.1 # 2,  23, 25
4.2 #2, 4, 5, 8
4.3 #2, 3, 6, 8, 11, 27, 28, 30
4.4 #6, 11, 33, 34
4.5 # 1, 3, 4, 7, 10, 16, 31, 3