Review for the FINAL EXAM I(Monday 12/13/04,  1:00-3:00PM).

Review Midterm I and midterm II.

CHAPTER 2 SYSTEMS - (Section 1-Section 4)

-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.
-Applications: Balancing Chemical Equations, 
-Review problems:
 2.1 # 16, 22, 24, 32
 2.2 # 8, 14, 24, 28, 41 
 2.3 # 4, 8, 12, 26, 42, 43, 44
 2.4 # 10, 12

CHAPTER 3 MATRICES (Section 1-Section5)

-Operations with matrices (properties).
-The transpose of a matrix (properties).
-The inverse of a matrix (properties).
-The fundamental theorem of Invertible Matrices (a)-(d).
-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
-FT of IM (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, 16, 22, 29, 37
 3.2 # 5, 14, 24, 44, 45, 47 
 3.3 # 4, 12, 22, 42, 45, 46, 52, 53
 3.4 # 2, 3, 16, 24, 35, 36, 39, 46, 51, 52 
 3.5 # 12, 21, 24, 36,  37

CHAPTER 4  EIGENVALUES and EIGENVECTORS - (Section 1-Section 4)
Section 1
-eigenvalues, eigenvectors, eigenspaces of a matrix
Section 2
-Determinants, Laplace Expansion Theorem
-Properties of determinants 
-determinants and matrix operations  
(from section4.2: Theorem 2, 3, 6, 7, 8, 9, 10)
-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. 
-FT of IM (version 3), Theorem 4, Theorem 5.
Section 4
-similar matrices, diagonalizable matrices.
- applications of Theorem 5 and Theorem 7 to finding  a diagonalization and  the powers of a matrix. 

-Review problems:
4.1 # 12, 23, 35 a)
4.2 # 26, 38, 46, 48, 49, 50, 52, 60
4.3 # 8,  15, 17, 18, 22
4.4 # 8, 10, 22, 24, 27

CHAPTER 5 ORTHOGONALITY (Section 1-Section4)
Section 1
-Orthogonal and orthonormal sets
-Coordinates in orthogonal and orthonormal bases (Theorem 2, 3)
-Orthogonal matrices- definition and properties
Section 2
-Orthogonal projections and orthogonal complements, Theorem 1, 3, 5.
Section 3
- The Gram-Schmidt Process and QR factorization 
section 4
-Know how to Orthogonally Diagonalize a matrix. Theorem 2, 3, 4 
-Review problems: 
 5.1 # 2, 8, 14, 18, 23, 24, 27
 5.2 # 2, 4, 12, 16, 20, 23 
 5.3 # 4, 5, 9, 15
 5.4 # 1, 4, 8, 11, 13, 14, 15, 1