LINEAR ALGEBRA by Shen, Li-Yong -- Read -- Imperial Library of Trantor

Index

Cover Title Copyright Contents Prefaces List of Figures Chapter 1 Vector Spaces

1.1 Vector Spaces 1.2 Linear Span and Linearly Independence 1.3 Bases for Vector Spaces 1.4 Linear System of Equations 1.5 A First Look of Determinants 1.6 Using Computer Algebra Systems to Perform Computations 1.7 Applications of Vector Spaces

1.7.1 Code 1.7.2 Hamming Code 1.7.3 Linear Code 1.7.4 Golay Code

1.8 Exercises

Chapter 2 Linear Transformations

2.1 Linear Transformations 2.2 Rank and Nullity of a Linear Transformation 2.3 Representation of Linear Transformations by Matrices 2.4 The Algebra of Linear Transformation 2.5 Applications of Linear Transformation

2.5.1 Affine Transformations 2.5.2 Projective Transformations

2.6 Exercises

Chapter 3 Linear Operators

3.1 Characteristic and Minimal Polynomials 3.2 Eigenspaces and Diagonalizability 3.3 Jordan Canonical Form 3.4 Trace 3.5 Determinants 3.6 Generalized Companion Matrix 3.7 Solution of Linear System of Equations 3.8 Some Applications of Matrices

3.8.1 Linear Programming 3.8.2 Geometric Transformation 3.8.3 Matrices in Graph Theory 3.8.4 Positional Relationship of Two Ellipsoids

3.9 Exercises

Chapter 4 Orthogonal Bases

4.1 Inner Product Spaces 4.2 Gram-Schmidt Process to Produce Orthonormal Basis 4.3 Orthogonal Complements and Projections 4.4 Orthogonal Projections and Reflections

4.4.1 Orthogonal Projection Matrix 4.4.2 Reflection Matrix

4.5 Properties of Symmetric Matrices 4.6 QR Factorization 4.7 Singular Value Decomposition 4.8 Applications of SVD and QR Decompositions

4.8.1 Image Processing 4.8.2 QR decomposition and GCD

4.9 Exercises

Chapter 5 Matrix Decomposition

5.1 Decomposition over R or C

5.1.1 Decompositions to Solve Linear Systems 5.1.2 Decompositions Based on Eigenvalues 5.1.3 Examples

5.2 Iterative Methods to Solve Linear Systems Numerically

5.2.1 Jacobi Iterative Method 5.2.2 Gauss-Seidel Iterative Method

5.3 Matrix over Principle Ideal Domain

5.3.1 Matrix Decomposition over PID

5.4 Exercises

Bibliography Index