PICT DRAFT Lecture Notes on Linear Algebra
PICT PICT DRAFT PICT PICT DRAFT PICT PICT DRAFT

Lecture Notes on Linear Algebra

Arbind K Lal
Sukant Pati

July 10, 2018
PICT PICT DRAFT
PICT PICT DRAFT
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 ℂn
 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 PICT PICT 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 W1 + W2
 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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