Notes on Mathematics - 102 (linear algebra)

Description

I Linear Algebra......Page 7 Definition of a Matrix......Page 9 Operations on Matrices......Page 10 Multiplication of Matrices......Page 12 Some More Special Matrices......Page 13 Block Matrices......Page 15 Matrices over Complex Numbers......Page 17 Introduction......Page 19 Definition and a Solution Method......Page 20 Row Operations and Equivalent Systems......Page 21 Gauss Elimination Method......Page 24 Row Reduced Echelon Form of a Matrix......Page 26 Gauss-Jordan Elimination......Page 27 Elementary Matrices......Page 29 Rank of a Matrix......Page 30 Example......Page 33 Main Theorem......Page 34 Inverse of a Matrix......Page 35 Equivalent conditions for Invertibility......Page 37 Inverse and Gauss-Jordan Method......Page 39 Determinant......Page 40 Adjoint of a Matrix......Page 43 Cramer's Rule......Page 45 Miscellaneous Exercises......Page 46 Definition......Page 49 Examples......Page 51 Subspaces......Page 53 Linear Combinations......Page 54 Linear Independence......Page 57 Bases......Page 58 Important Results......Page 60 Ordered Bases......Page 66 Definitions and Basic Properties......Page 69 Matrix of a linear transformation......Page 72 Rank-Nullity Theorem......Page 75 Similarity of Matrices......Page 80 Definition and Basic Properties......Page 87 Gram-Schmidt Orthogonalisation Process......Page 92 Orthogonal Projections and Applications......Page 100 Matrix of the Orthogonal Projection......Page 103 Introduction and Definitions......Page 107 diagonalization......Page 113 Diagonalizable matrices......Page 116 Sylvester's Law of Inertia and Applications......Page 121 II Ordinary Differential Equation......Page 129 Introduction and Preliminaries......Page 131 Equations Reducible to Separable Form......Page 134 Exact Equations......Page 136 Integrating Factors......Page 138 Linear Equations......Page 141 Miscellaneous Remarks......Page 143 Initial Value Problems......Page 145 Orthogonal Trajectories......Page 149 Numerical Methods......Page 150 Introduction......Page 153 Wronskian......Page 156 Method of Reduction of Order......Page 159 Second Order equations with Constant Coefficients......Page 160 Non Homogeneous Equations......Page 162 Variation of Parameters......Page 164 Higher Order Equations with Constant Coefficients......Page 166 Method of Undetermined Coefficients......Page 170 Introduction......Page 175 Properties of Power Series......Page 177 Solutions in terms of Power Series......Page 179 Statement of Frobenius Theorem for Regular (Ordinary) Point......Page 180 Introduction......Page 181 Legendre Polynomials......Page 182 III Laplace Transform......Page 189 Definitions and Examples......Page 191 Examples......Page 192 Properties of Laplace Transform......Page 194 Transform of Unit Step Function......Page 199 Limiting Theorems......Page 200 Application to Differential Equations......Page 202 Transform of the Unit-Impulse Function......Page 204 IV Numerical Applications......Page 207 Forward Difference Operator......Page 209 Backward Difference Operator......Page 211 Central Difference Operator......Page 213 Relations between Difference operators......Page 214 Newton's Interpolation Formulae......Page 215 Divided Differences......Page 221 Lagrange's Interpolation formula......Page 224 Gauss's and Stirling's Formulas......Page 226 Numerical Differentiation......Page 229 A General Quadrature Formula......Page 233 Trapezoidal Rule......Page 234 Simpson's Rule......Page 235 System of Linear Equations......Page 239 Determinant......Page 242 Properties of Determinant......Page 246 Dimension of M + N......Page 250 Proof of Rank-Nullity Theorem......Page 251 Condition for Exactness......Page 252