Table of Contents
Chapter 1 Equation Systems and Matrices
1．1 Systems of Linear Equations
1．1．1 Brief History of Algebra and Linear Algebra
1．1．2 Systems of Linear Equations
1．1．3 Strict Triangular Form of Linear Systems
1．2 Linear System in Matrix
1．2．1 Matrix Notations
1．2．2 Solving Linear Systems
1．3 Reduced Row Echelon Form
1．3．1 Row Echelon Form
1．3．2 Gauss Elimination
1．3．3 Reduced Row Echelon
1．4 Consistency of Linear Systems
1．4．1 Overdetermined Systems
1．4．2 Underdetermined Systems
1．4．3 Homogeneous Systems
Chapter 2 Matrix Algebra
2．1 Notations and Operations
2．1．1 Matrix Notations
2．1．2 Matrix Operations
2．1．3 Algebraic Rules of Matrix Operations
2．2 Inverse and Transpose of Matrices
2．2．1 Identity Matrix
2．2．2 Matrix Inverse
2．2．3 The Transpose of a Matrix
2．2．4 Triangular and Diagonal Matrices
2．3 Partitioned Matrices
2．3．1 The Notations of Partitioned Matrices
2．3．2 Block Addition and Scalar Multiplication
2．3．3 Block Multiplication
2．4 Linear Combination of Vectors
2．4．1 Linear Combination of Vectors
2．4．2 Equivalent Systems
2．4．3 Elementary Matrices
2．4．4 Find the Inverse Matrix
2．5 The Determinant of a Matrix
2．5．1 CASE I The Determinant of 1 x 1 Matrices
2．5．2 CASE II The Determinant of 2 x 2 Matrices
2．5．3 CASE III 3 x 3 Matrices
2．5．4 CASE IV The Determinant of n x n Matrices
2．6 Properties of Determinants
2．6．1 Determinant of the Transposed Matrix
2．6．2 Determinant of Triangular Matrices
2．6．3 Determinant of Matrices with All Zeros in a Row or Column
2．6．4 Determinant of Matrices with Identical Rows or Columns
2．6．5 *Laplace's Definition of Determinant by Using Subdeterminant
2．6．6 Algebraic Rules of Determinants
2．6．7 Determinant and Singularity of a Matrix
2．7 Cramer's Rule
2．7．1 The Adjoint of a Matrix
2．7．2 Cramer's Rule
Chapter 3 Vector Spaces
3．1 Definitions and Examples
3．1．1 Definitions
3．1．2 Examples
3．1．3 Euclidean Vector Space
3．1．4 Inner Product and Outer Product Expansion of Vectors
3．2 Subspaces
3．2．1 Definitions
3．2．2 The Null Space of a Matrix
3．2．3 The Span of Vectors
3．3 Linear Independence
3．3．1 Concepts and Examples
3．3．2 The Minimal Spanning Set of a Vector Space
3．3．3 The Minimal Spanning Set of Nullspace of a Matrix
3．4 Basis and Dimension
3．4．1 Basis of Vector Spaces
3．4．2 Dimension of Vector Spaces
3．5 Changing of Basis
3．5．1 Coordinate of Vector
3．5．2 Changing of Basis in R2
3．5．3 Changing of Basis in an n-dimensional Vector Space
3．6 Row Space and Column Space of Matrices
3．6．1 Concepts and Examples
3．6．2 Rank of a Matrix
3．6．3 The Rank and Nullity Theorem
Chapter 4 Analytic Geometry
4．1 Analytic Geometry and Cartesian Coordinate System
4．1．1 Cartesian Coordinate System on Plane
4．1．2 Cartesian Coordinate System in Space
4．1．3 Vectors in Cartesian Coordinate System
4．2 Algebra in Euclidean Geometry
4．2．1 Euclidean Length
4．2．2 Included Angle of Two Vectors
4．2．3 The Geometric Interpretations of Operations on Vectors'
4．2．4 The Projection of Vectors
4．2．5 Inner Product
4．2．6 Cross Product
4．2．7 The Triple Scalar or Box Product
4．3 Planes and Lines
4．3．1 The Equation and Figure of Space Surface
4．3．2 The Equation of a Plane
4．3．3 The Relative Positions of Planes
4．3．4 The Equation of a Line
4．3．5 The Relative Positions of Lines
4．3．6 The Relative Positions Between a Line and a Plane
4．3．7 The Distance from a Point to a Plane or a Line
Chapter 5 Linear Transformation
5．1 Definition and Examples
5．2 The Image and Kernel
5．3 Matrix Representation of Linear Transformations
5．4 Similar Matrices
Chapter 6 Matrix Diagonalization
6．1 Inner Product and Inner Product Space
6．2 Orthonormal Sets and Orthogonal Subspaces
6．2．1 Orthonormal Sets
6．2．2 Orthogonal Matrices
6．2．3 Orthogonal Subspaces
6．3 The Gram-Schmidt Orthogonalization Process
6．4 Eigenvalues and Eigenvectors
6．4．1 Concepts and Examples
6．4．2 The Product and Sum of the Eigenvalues
6．4．3 The Eigenvalues and Eigenvectors of Similar Matrices
6．5 Diagonalization
Chapter 7 Quadratic Form and Its Applications
7．1 Quadratic Form and Its Matrix Representation
7．2 The Diagonalization of Real Symmetric Matrices
7．3 Conic Sections and Quadric Surfaces
7．3．1 Conic Sections
7．3．2 Quadric Surfaces
Bibliography
