Notes on Mathematics - 102 (linear algebra)

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