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## Lecture Notes on Linear Algebra

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July 10, 2018

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Contents

1 Introduction to Matrices

1.1 Definition of a Matrix

1.2 Operations on Matrices

1.3 Some More Special Matrices

1.4 Summary

2 System of Linear Equations

2.1 Introduction

2.2 Main Ideas of Linear Systems

2.3 Square Matrices and Linear Systems

2.4 Miscellaneous Exercises

2.5 Summary

3 Vector Spaces

3.1 Vector Spaces: Definition and Examples

3.2 Fundamental Subspaces Associated with a Matrix

3.3 Linear Independence

3.4 Basis of a Vector Space

3.5 Application to the subspaces of ℂ^{n}

3.6 Ordered Bases

3.7 Summary

4 Linear Transformations

4.1 Definitions and Basic Properties

4.2 Rank-Nullity Theorem

4.3 Matrix of a linear transformation

4.4 Similarity of Matrices

4.5 Dual Space*

4.6 Summary

5 Inner Product Spaces

5.1 Definition and Basic Properties

5.2 Gram-Schmidt Orthonormalization Process

5.3 Orthogonal Projections and Applications

5.4 Orthogonal Operator and Rigid Motion

5.5 Summary

6 Eigenvalues, Eigenvectors and Diagonalizability

6.1 Introduction and Definitions DRAFT

6.2 Diagonalization

6.3 Quadratic Forms

7 Jordan Canonical form

7.1 Jordan Canonical form theorem

7.2 Minimal polynomial

7.3 Applications of Jordan Canonical Form

8 Advanced Topics on Diagonalizability and Triangularization^{*}

8.1 More on the Spectrum of a Matrix

8.2 Methods for Tridiagonalization and Diagonalization

8.3 Commuting Matrices and Simultaneous Diagonalization

9 Appendix

9.1 Uniqueness of RREF

9.2 Permutation/Symmetric Groups

9.3 Properties of Determinant

9.4 Dimension of W_{1} + W_{2}

9.5 When does Norm imply Inner Product

9.6 Roots of a Polynomials

9.7 Variational characterizations of Hermitian Matrices

Index

Index

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1 Introduction to Matrices

1.1 Definition of a Matrix

1.2 Operations on Matrices

1.3 Some More Special Matrices

1.4 Summary

2 System of Linear Equations

2.1 Introduction

2.2 Main Ideas of Linear Systems

2.3 Square Matrices and Linear Systems

2.4 Miscellaneous Exercises

2.5 Summary

3 Vector Spaces

3.1 Vector Spaces: Definition and Examples

3.2 Fundamental Subspaces Associated with a Matrix

3.3 Linear Independence

3.4 Basis of a Vector Space

3.5 Application to the subspaces of ℂ

3.6 Ordered Bases

3.7 Summary

4 Linear Transformations

4.1 Definitions and Basic Properties

4.2 Rank-Nullity Theorem

4.3 Matrix of a linear transformation

4.4 Similarity of Matrices

4.5 Dual Space*

4.6 Summary

5 Inner Product Spaces

5.1 Definition and Basic Properties

5.2 Gram-Schmidt Orthonormalization Process

5.3 Orthogonal Projections and Applications

5.4 Orthogonal Operator and Rigid Motion

5.5 Summary

6 Eigenvalues, Eigenvectors and Diagonalizability

6.1 Introduction and Definitions DRAFT

6.2 Diagonalization

6.3 Quadratic Forms

7 Jordan Canonical form

7.1 Jordan Canonical form theorem

7.2 Minimal polynomial

7.3 Applications of Jordan Canonical Form

8 Advanced Topics on Diagonalizability and Triangularization

8.1 More on the Spectrum of a Matrix

8.2 Methods for Tridiagonalization and Diagonalization

8.3 Commuting Matrices and Simultaneous Diagonalization

9 Appendix

9.1 Uniqueness of RREF

9.2 Permutation/Symmetric Groups

9.3 Properties of Determinant

9.4 Dimension of W

9.5 When does Norm imply Inner Product

9.6 Roots of a Polynomials

9.7 Variational characterizations of Hermitian Matrices

Index

Index