Contents

• Linear Algebra
  • Matrices
    • Definition of a Matrix
      • Special Matrices
    • Operations on Matrices
      • Multiplication of Matrices
      • Inverse of a Matrix
    • Some More Special Matrices
      • Submatrix of a Matrix
      • Block Matrices
    • Matrices over Complex Numbers
  • Linear System of Equations
    • Introduction
      • A Solution Method
    • Row Operations and Equivalent Systems
      • Gauss Elimination Method
    • Row Reduced Echelon Form of a Matrix
      • Gauss-Jordan Elimination
      • Elementary Matrices
    • Rank of a Matrix
    • Existence of Solution of $A {\mathbf x}= {\mathbf b}$
      • Example
      • Main Theorem
      • Equivalent conditions for Invertibility
      • Inverse and the Gauss-Jordan Method
    • Determinant
      • Adjoint of a Matrix
      • Cramer's Rule
    • Miscellaneous Exercises
  • Finite Dimensional Vector Spaces
    • Vector Spaces
      • Definition
      • Examples
      • Subspaces
      • Linear Combinations
    • Linear Independence
    • Bases
      • Important Results
    • Ordered Bases
  • Linear Transformations
    • Definitions and Basic Properties
    • Matrix of a linear transformation
    • Rank-Nullity Theorem
    • Similarity of Matrices
  • Inner Product Spaces
    • Definition and Basic Properties
    • Gram-Schmidt Orthogonalisation Process
    • Orthogonal Projections and Applications
      • Matrix of the Orthogonal Projection
  • Eigenvalues, Eigenvectors and Diagonalisation
    • Introduction and Definitions
    • Diagonalisation
    • Diagonalisable matrices
    • Sylvester's Law of Inertia and Applications
  
  
• Ordinary Differential Equation
  • Differential Equations
    • Introduction and Preliminaries
    • Separable Equations
      • Equations Reducible to Separable Form
    • Exact Equations
      • Integrating Factors
    • Linear Equations
    • Miscellaneous Remarks
    • Initial Value Problems
      • Orthogonal Trajectories
    • Numerical Methods
  • Second Order and Higher Order Equations
    • Introduction
    • More on Second Order Equations
      • Wronskian
      • Method of Reduction of Order
    • Second Order equations with Constant Coefficients
    • Non Homogeneous Equations
    • Variation of Parameters
    • Higher Order Equations with Constant Coefficients
    • Method of Undetermined Coefficients
  • Solutions Based on Power Series
    • Introduction
      • Properties of Power Series
    • Solutions in terms of Power Series
    • Statement of Frobenius Theorem for Regular (Ordinary) Point
    • Legendre Equations and Legendre Polynomials
      • Introduction
      • Legendre Polynomials
  
  
• Laplace Transform
  • Laplace Transform
    • Introduction
    • Definitions and Examples
      • Examples
    • Properties of Laplace Transform
      • Inverse Transforms of Rational Functions
      • Transform of Unit Step Function
    • Some Useful Results
      • Limiting Theorems
    • Application to Differential Equations
    • Transform of the Unit-Impulse Function
  
  
• Numerical Applications
  • Newton's Interpolation Formulae
    • Introduction
    • Difference Operator
      • Forward Difference Operator
      • Backward Difference Operator
      • Central Difference Operator
      • Shift Operator
      • Averaging Operator
    • Relations between Difference operators
    • Newton's Interpolation Formulae
  • Lagrange's Interpolation Formula
    • Introduction
    • Divided Differences
    • Lagrange's Interpolation formula
    • Gauss's and Stirling's Formulas
  • Numerical Differentiation and Integration
    • Introduction
    • Numerical Differentiation
    • Numerical Integration
      • A General Quadrature Formula
      • Trapezoidal Rule
      • Simpson's Rule
  • Numerical Methods
    • Introduction
      • Euler's Method
    • Error Estimates and Convergence
    • Runge-Kutta Method
      • Algorithm: Runge-Kutta Method of Order $2$
      • Runge-Kutta Method of Order $4$
    • Predictor-Corrector Methods
      • Algorithm for Predictor-Corrector Method
  • Appendix
    • System of Linear Equations
    • Determinant
    • Properties of Determinant
    • Dimension of $M + N$
    • Proof of Rank-Nullity Theorem
    • Condition for Exactness