Prof. Arun Sharma‘s ****Course Notes:

MATH54_NOTE_LinearAlgebra.pdf

MATH54_NOTE_DifferentialEquations.pdf

The Art of Linear Algebra:

The-Art-of-Linear-Algebra.pdf

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Contents

(i) Linear Alegebra

1. Linear Equations in Linear Algebra

• 1.1 Systems of Linear Equations
• 1.2 Row Reduction and Echelon Forms
• 1.3 Vector Equations
• 1.4 The Matrix Equation Ax=b
• 1.5 Solution Sets of Linear Systems
• 1.7 Linear Independence
• 1.8 Introduction to Linear Transformations
• 1.9 The Matrix of a Linear Transformation

2. Matrix Algebra

• 2.1 Matrix Operations
• 2.2 The Inverse of a Matrix
• 2.3 Characterization of Invertible Matrices (Invertible Matrix Theorem)

3. Determinants

• 3.1 Introduction to Determinants
• 3.2 Properties of Determinants
• 3.3 Cramer’s Rule

4. Vector Spaces

• 4.1 Vector Spaces and Subspaces
• 4.2 Null Spaces, Column Spaces, and Linear Transformations
• 4.3 Linearly Independent Sets; Bases
• 4.4 Coordinate Systems
• 4.5 The Dimension of a Vector Space
• 4.6 Rank
• 4.7 Change of Basis

5. Eigenvalues and Eigenvectors

• 5.1 Eigenvectors and Eigenvalues
• 5.2 The Characteristic Equation
• 5.3 Diagonalization
• 5.4 Eigenvectors and Linear Transformations
• 5.5 Nonreal Eigenvalues

6. Orthogonality and Least Squares

• 6.1 Inner Product, Length, and Orthogonality
• 6.2 Orthogonal Sets
• 6.3 Orthogonal Projections
• 6.4 The Gram-Schmidt Procedure
• 6.5 Least-Squares Problems
• 6.6 Least Squares Curve / Design Matrix
• 6.7 Inner Product Space / Cauchy-Schwarz Inequality

7. Symmetric Matrices

• 7.1 Diagonalization of Symmetric Matrices
• 7.3 Constrained Optimization
• 7.4 The Singular Value Decomposition
• 7.5 Applications to Image Processing and Statistics

(ii) Differential Equations

4. Linear Second-Order Equations

• 4.2 Homogeneous Linear Equations: The General Solution
• 4.3 Auxiliary Equations with Complex Roots
• 4.4 Non-homogeneous Equations: the Method of Undetermined Coefficients
• 4.5 The Superposition Principle and Undetermined Coefficients Revisited
• 4.6 Variation of Parameters

9. Matrix Methods for Linear Systems

• 9.1 Introduction
• 9.4 Linear Systems in Normal Form
• 9.5 Homogeneous Linear Systems with Constant Coefficients
• 9.6 Complex Eigenvalues
• 9.7 Nonhomogeneous Linear Systems