Review for the FINAL EXAM I(Tuesday 12/14/10,  1:00-3:00PM).

Review Midterm I Midterm II and the solution to Homework/quzess  sets.

CHAPTER 2 SYSTEMS - (Sections 2.1-2.3)

-Consistent (inconsistent) systems. 
-Augmented matrix.
-Row echelon form - Gaussian Elimination
-Reduced row echelon form- Gauss-Jordan Elimination.
-The rank Theorem.
-Homogeneous systems.
-Span, spanning sets.
-Linearly dependent/independent vectors and
 the connection with homogeneous systems theory.
-Know how to find the inverse of a 2 by 2 or 3 by 3 matrix.
-subspaces,  basis,  dimension
-know to find bases for the row space, the column space 
  and the  null space of a matrix
-rank, Rank Theorem
-FTIM (version 1 & 2)
-coordinate of a vector with respect to a basis
Section  6
-Linear transformations
-The matrix associated with a linear transformation
-Composition of linear transformations

-Review problems:
 2.1 # 22, 24 26, 38
 2.2 # 8, 14, 24, 26, 28, 36, 38, 41, 42.
 2.3 # 3, 8, 12, 10, 20, 24, 28, 34, 42, 43,  44, 46, 48


CHAPTER 3 MATRICES (Sections 3.1-3.3, 3.5  and Section 3.6))

-Operations with matrices (properties).
-The transpose of a matrix (properties).
-The inverse of a matrix (properties).
-Symmetric matrices, properties. 
-Know how to find the inverse of a 2 by 2 or 3 by 3 matrix.
-subspaces,  basis,  dimension
-know to find bases for the row space, the column space 
  and the  null space of a matrix
-rank, Rank Theorem, FTIM (version 2)
-coordinate of a vector with respect to a basis
Section  5
-Linear transformations
-The matrix associated with a linear transformation
-Composition of linear transformations

-Review problems: 
 3.1 # 12, 13, 29, 34, 36, 37, 38
 3.2 # 1, 3, 22, 23, 27, 35, 44, 45, 47
 3.3 # 3, 4, 9, 12, 17, 21, 22, 33, 42, 44, 45, 46, 47, 49, 53, 64, 65
 3.5 # 55, 56, 58, 63
 3.6 # 23, 36, 39


CHAPTER 4  EIGENVALUES and EIGENVECTORS -  (Sections 4.1-4.4)
Section 1
-Definitions for eigenvalues, eigenvectors, eigenspaces of a matrix
-Know how to find  eigenvalues, eigenvectors, eigenspaces for a 2 by 2   or  3 by 3 matrix
Section 2
-Determinants, (Laplace) Expansion Theorem
-Properties of Determinants 
(from section 4.2: Theorem 2, 3, 6, 7, 8, 9, 10)
-The  formula for the inverse of a matrix  (Theorem 12)
-Cramers's Rule 
Section 3
- Know to find the characteristic polynomial, the  characteristic equation and bases for eigenspaces of a matrix.
-eigenvalues of a triangular matrix. 
-FTIM (version 3), Theorem 15-20.
Section 4
-similar matrices, diagonalizable matrices.  Theorem 21-27
-know how to diagonalize a matrix  and how to find  the powers of a matrix. 


4.1 # 25, 35, 37
4.2 # 1, 3, 7, 9, 17, 25, 35, 39, 49, 53, 55
4.2 # 4, 8, 12, 14, 26, 28, 36, 38, 40, 46, 50, 52, 54, 56, 58 66
4.3 #  3, 4, 8, 13, 15, 17, 18, 19, 20, 21, 22 
4.4  # 1, 5, 7, 11, 25, 31, 35, 41, 45, 10, 16, 22, 24, 30, 32, 34, 36, 42    

CHAPTER 5 ORTHOGONALITY (Section 5.1-Section 5.4)
Section 1
-Orthogonal and orthonormal sets
-Coordinates in orthogonal and orthonormal bases (Theorem 1-8)
-Orthogonal matrices- definition and properties
Section 2
-Orthogonal projections and orthogonal complements, Theorem 9, 10, 11, 13.
Section 3
- The Gram-Schmidt Process and QR factorization 
section 4
-Know how to Orthogonally Diagonalize a matrix. Theorem 17, 18, 19, 20 
-Review problems: 
 5.1 # # 3, 5, 11, 17, 23, 27, 33,  8, 18, 12, 14, 18, 22, 24
 5.2 #  2, 4, 12, 16, 18, 20, 7, 19, 23, 25 
 5.3 # 3, 5, 9, 15, 17, 20.    
 5.4 # 3, 7, 11, 13, 14, 15, 16, 21, 23.