Linear Algebra

Author:              Zhiming Ou

                         Mattermatics Learning Center

Publisher:       3265 Public Way

ISBN: 978-1-0677109-1-0

Summary

Linear algebra is a branch of mathematics that uses vectors, matrices, and linear transformations to study the linear structure of spaces with finite dimensions, both algebraically and geometrically. Linear means that, the variables are all in degree 1.

The important achievements include (1) Elimination can be carried out by a matrix, which can be done step-by-step, so can be coded, (2) Determinants measure the size of a matrix, so the content of regions in any space; (3) Inner products allow the measurements of angles between any two vectors; (4) quadratic forms can be expressed as matrices, so as to determine the extreme values for functions with a finite number of variables.

There should be a generalization to spaces with infinite dimensions, that is Hilbert spaces. The behavior of functions with countably many variables is studied in Functional Analysis, one of the 6 analyses courses in mathematics. The means for solving equations of infinitely many variables is still a challenge for mathematicians.

Content

Chapter 1 Systems of Linear Equations     4

  • 1.1 2 × 2 Systems of Linear Equations 5

Gaussian Elimination

Determinants with the Second Order    6

  • 1.2 3 × 3 Systems of Linear Equations 7
  • 1.3 m × n Systems of Linear Equations 10
  • 1.4 Structure of Solution 12

 

Chapter 2 Matrices    17

  • 2.1 Concepts 17
  • 2.2 Operations with Matrices 18
  • 2.3 Elementary Operations and Elementary Matrices 23
  • 2.4 Inverse of a Square Matrix 25
  • 2.5 Special Types of Square Matrices 29
  • 2.6 Block Matrices 31
  • 2.7 LU Factorization 33

 

Chapter 3 Determinants    36

  • 3.1 Determinants of Order 1, 2, and 3 36
  • 3.2 Permutations and their Inversion Number 37
  • 3.3 Determinants of any Order 39
  • 3.4 Properties of Determinants 40
  • 3.5 Evaluations of Determinants 45
  • 3.6 Applications of Determinants 46

 

Chapter 4 Vectors and Vector Space    50

  • 4.1 Vectors in R2 and R3
  • 4.2 Vectors in Rn 53
  • 4.3 Vector Spaces 55
  • 4.4 Linear Combinations and Spans 60
  • 4.5 Linear Independence, Basis, and Dimension 64
  • 4.6 Coordinates and Change of Basis 73

 

Chapter 5 Inner Product Space       79

5.1 Inner Product Space     79

5.2 Examples of Inner Product Spaces      80

5.3 Cauchy-Schwarz Inequality    81

5.4 Angle between Vectors     81

5.5 Orthogonality     82

5.6 Orthogonal Sets and Bases   84

5.7 Projections       86

5.8 Gram-Schmidt Orthogonalization Process

  1. 9 Orthogonal Matrices 89

5.10 Normed Vector Space      91

 

Chapter 6 Linear Transformation     93

  • 6.1 Mappings and Functions 93
  • 6.2 Linear Mappings (Transformations) 96
  • 6.3 Operations with Linear Mapping 103
  • 6.4 Algebra A(V) of Linear Operators 104
  • 6.5 Linear Mappings and Matrix Representations 107

 

Chapter 7 Diagonalization      115

Polynomials of Matrices   116

Characteristic Polynomial      116

Eigenvalues and eigenvectors   119

Diagonalization of Linear Operators   121

Diagonalizing Matrices      123

Minimal Polynomial     125

Jordan Normal Form      128

 

Chapter 8 Quadratic Forms    131

  • 8.1 Concepts 131
  • 8.2 Congruent Symmetric Matrices 133
  • 3 Orthogonal Diagonalization 136
  • 8.4 Positive Definite Matrices and Quadratic Forms 138