Next:
Contents
Contents
Notes on Mathematics - 102
0
1
Peeyush Chandra,
A. K. Lal,
V. Raghavendra,
G. Santhanam
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
Miscellaneous Exercises
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
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
Runge-Kutta Method of Order
Predictor-Corrector Methods
Algorithm for Predictor-Corrector Method
Appendix
System of Linear Equations
Determinant
Properties of Determinant
Dimension of
Proof of Rank-Nullity Theorem
Condition for Exactness
About this document ...
S Ghorai 2007-09-12
A. K. Lal,
V. Raghavendra,
G. Santhanam
