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Mathematical Method Book by S M Yousuf

## Content

PREFACE

CHAPTER 1

TRIGONOMETRY 1

Section I

Complex Numbers 1

The Argand’s Diagram 7

EXERCISE 1.1 20

De Moivre’s Theorem 21

Roots of a Complex Number 31

EXERCISE 1.2 35

Basic Elementary Functions 37

EXERCISE 1.3 41

Logarithmic Function 43

Inverse Hyperbolic Functions 45

Inverse Trigonometric Functions48

Complex Powers 50

EXERCISE 1.4 53

Summation of Series 54

EXERCISE 1.5 59

CHAPTER 2

GROUPS 61

Definition and Examples

Properties of Groups 65

EXERCISE 2.1 69

Subgroups 71

Cyclic Groups 75

Cosets – Lagrange’s Theorem 77

EXERCISE 2.2 80

Permutations 82

Cycles 87

Transpositions 90

Order of a Permutation

EXERCISE 2.3

Rings and Fields 98

EXERCISE 2.4 105

CHAPTER 3

MATRICES 120

Introduction 107

Algebra of Matrices 110

Partitioning of Matrices

EXERCISE 3.1 126

Inverse of a Matrix 130

Elementary Row Operations

Elementary Column Operations

EXERCISE 3.2 144

CHAPTER 4 SYSTEMS OF LINEAR EQUATIONS 149

Preliminaries 149

Equivalent Fractions 150

CHAPTER 4 SYSTEMS OF LINEAR EQUATIONS 149

Preliminaries 149

Equivalent Equations 150

Gaussian Elimination Method 152

Gauss-Jordan Elimination Method 158

Consistency Criterion

Network Flow Problems 170

EXERCISE 4 174

CHAPTER 5 DETERMINANTS

Determinant of a Square Matrix

Axiomatic Definition of a Determinant 185

Determinant as Sum of Products of Elements

Determinant of the Transpose

An Algorithm to Evaluate Det A 195

Determinants and Inverses of Matrices

EXERCISE 5.1 209

Miscellaneous Results 216

EXERCISE 5.2 219

CHAPTER 6

VECTOR SPACES

223

Definition and Examples 223

Subspaces 230

Linear Combinations and Spanning Sets

233

EXERCISE 6.1 236

Linear Dependence and Basis 239

EXERCISE 6.2 251

Row and Column Spaces 253

Rank-Alternative Method 255

Proofs of Theorems in Chapter 4

Linear Transformations 264

EXERCISE 6.3 272

Matrix of a Linear Transformation 274

EXERCISE 6.4 279

CHAPTER 7

INNER PRODUCT SPACES 281

Definitions and Examples 281

Orthogonality 286

EXERCISE 7.1 294

Orthogonal Matrices 295

EXERCISE 7.2 299

Eigenvalues and Eigenvectors

EXERCISE 7.3 306

Similar Matrices 307

Symmetric Matrices 310

Diagonalization of Matrices

EXERCISE 7.4 323

CHAPTER 8

INFINITE SERIES

Sequences 325

EXERCISE 8.1 330

Infinite Series 331

The Basis Comparison Test 339

The Limit Comparison Test

The Integral Test 343

EXERCISE 8.2 347

The Ratio Test 349

Cauchy’s Root Test 352

EXERCISE 8.3 354

Alternating Series 356

Absolute and Conditional Convergence 358

EXERCISE 8.4 362

Power Series 363

Convergence of Power Series 364

Differentiation and Integration of Power Series 370

Addition and Multiplication of Power Series 373

EXERCISE 8.5 375

CHAPTER 9

FIRST-ORDER DIFFERENTIAL

EQUATIONS 377

D.E. and Their Classifications377

Formation of Differential Equations 380

Initial and Boundary Conditions 381

EXERCISE 9.1 384

Separable Equations 386

EXERCISE 9.2 388

Homogeneous Equations 388

D.E. Reducible to Homogeneous Form

EXERCISE 9.3 393

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1. Mohsin Raza says:

Thanks for providing this book free of cost

2. Hanan says:

3. Shahzaib Ali says:

I am thankful, you have provided full book.
thanks again. Allah pak slamat rakhy

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