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# Notes of Mathematical Method

Notes of the Mathematical Method written by by S.M. Yusuf, A. Majeed and M. Amin and published by Ilmi Kitab Khana, Lahore.

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## Chapter 01: Complex Numbers

A complex number is an element $(x,y)$ of the set $$ \mathbb{R}^2=\{(x,y): x,y \in \mathbb{R}\} $$ obeying the following rules of addition and multiplication.

For $z_1=(x_1,y_1)$, $z_2=(x_2,y_2)$, we put

- $z_1+z_2= (x_1+x_2, y_1+y_2)$
- $z_1 z_2 = (x_1 x_2 - y_1 y_2, x_1 y_2+y_1 x_2)$

The set $\mathbb{R}^2$ with operation defined above is denoted by $\mathbb{C}$.

- Complex numbers
- Properties of complex numbers
- The Argand's diagram
- De Moivre's theorem
- Roots of the complex numbers
- Basic elementary functions
- Logarithmic functions
- Inverse hyperbolic functions
- Inverse trigonometric functions
- Complex power
- Summation of series

Exercise 1.1 | Download PDF (470.64 KiB, 29538 downloads) | View online |

Exercise 1.2 | Download PDF (1.27 MiB, 18683 downloads) | View online |

Exercise 1.3 | Download PDF (1.64 MiB, 14685 downloads) | View online |

Exercise 1.4 | Download PDF (653.33 KiB, 28674 downloads) | View online |

Exercise 1.5 | Download PDF (690.22 KiB, 26551 downloads) | View online |

## Chapter 02: Groups

- Definition (axioms of group)
- Definition ( commutative group )
- Definition (idempotent)
- Properties of Group
- Theorem (The Cancellation Law)
- Theorem (Solution of Linear Equations )
- Subgroups
- Definition ( subgroup )
- Cyclic Groups
- Definition ( cyclic group )
- Cosets-Lagrange’s Theorem
- Permutations
- Cycles
- Transpositions
- Order of a Permutation
- Rings and Fields
- Properties of Rings

Notes by Shahzad Ahmed Khan: | ||
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Exercise 2.1 | Download PDF (606 KiB, 5299 downloads) | View online |

Exercise 2.2 | Download PDF (602.28 KiB, 3223 downloads) | View online |

Exercise 2.3 | Download PDF (657.81 KiB, 3871 downloads) | View online |

Notes by Shariq Mehtab Syed: | ||
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Exercise 2.1 | Download PDF (774.94 KiB, 20763 downloads) | View online |

Exercise 2.2 | Download PDF (977.55 KiB, 22373 downloads) | View online |

Exercise 2.3 | Download PDF (410.76 KiB, 15902 downloads) | View online |

## Chapter 03: Matrices

The difficulty level of this chapter is very low. Most of the questions involve calculations. This chapter is wide range of applications in Linear Algebra. In many universities teachers include this chapter in the syllabus of Linear Algebra for BS students of mathematics and other subjects.

- Introduction
- Algebra of matrices
- Partitioning of matrices
- Inverse of a matrix
- Elementary row operations
- Elementary column operations

Articles of Ex 3.1 | Download PDF (327.93 KiB, 14300 downloads) | View online |

Exercise 3.1 | Download PDF (630.02 KiB, 28606 downloads) | View online |

Exercise 3.2 | Download PDF (781.02 KiB, 31860 downloads) | View online |

## Chapter 04: System of Linear Equations

The difficulty level of this chapter is low. Most of the questions involve calculations. This chapter is wide range of applications in Linear Algebra and Operations Research. In many universities teachers include this chapter in the syllabus of Linear Algebra and Operations Research for BS students of mathematics and other subjects.

- Preliminaries
- Equivalent equations
- Gaussian elimination method
- Gauss-Jordan elimination method
- Consistency criterion
- Network flow problems

Exercise 4 | Download PDF) (1 MiB, 29212 downloads) | View online |

## Chapter 05: Determinants

- Determinant of a square matrix
- Axiomatic definition of a determinant
- Determinant as sum of products of elements
- Determinant of the transpose
- An algorithm to evaluate Det A
- Determinants and inverse of matrices

Exercise 5.1 | Download PDF (1.27 MiB, 17472 downloads) | View online |

Exercise 5.2 | Download PDF (395.67 KiB, 27887 downloads) | View online |

## Chapter 06: Vector Spaces

- Subspaces
- Linear combinations and spanning sets
- Linear dependence and basis
- Row and column spaces
- Rand-Alternative method
- Linear transformations
- Matrix of linear transformation

Exercise 6.1 | Download PDF (3.52 MiB, 10040 downloads) | View online |

Exercise 6.2 | Download PDF (1.23 MiB, 253 downloads) | View online |

Exercise 6.3 | Download PDF (861.66 KiB, 25911 downloads) | View online |

## Chapter 07: Inner Product Spaces

Inner product spaces form and important topic of Functional Analysis. These are simply vector space over the field of real or complex numbers and with an inner product defined on them.

- Definition and examples
- Orthogonality
- Orthogonal matrices
- Eigenvalues and Eigenvectors
- Similar matrices
- Symmetric matrices
- Diagonalization of matrices

Articles of Exercise 7.1 | Download PDF (510.1 KiB, 2511 downloads) | View online |

Solutions of Exercise 7.1 | Download PDF (363.36 KiB, 2792 downloads) | View online |

Articles of Exercise 7.2 | Download PDF (131.18 KiB, 1190 downloads) | View online |

Solutions of Exercise 7.2 | Download PDF (111.31 KiB, 1310 downloads) | View online |

Articles of Exercise 7.3 | Download PDF (157.9 KiB, 1261 downloads) | View online |

Solutions of Exercise 7.3 | Download PDF (519.81 KiB, 3598 downloads) | View online |

## Chapter 08: Infinite Series

Infinite series are of great importance in both pure and applied mathematics. They play a significant role in Physics and engineering. In fact many functions can be represented by infinite series. The theory of infinite series is developed through the use of special kind of function called sequence.

- Sequences
- Infinite series
- The basic comparison test
- The limit comparison test
- The integral test
- The ratio test
- Cauchy's root test
- Alternating series
- Absolute

Exercise 8.1 | Download PDF (494.55 KiB, 7230 downloads) | View online |

Exercise 8.2 | Download PDF (1.09 MiB, 10993 downloads) | View online |

Exercise 8.3 | Download PDF (658.47 KiB, 9008 downloads) | View online |

Exercise 8.4 | Download PDF (854.59 KiB, 8431 downloads) | View online |

Exercise 8.5 | Download PDF (1.46 MiB, 10590 downloads) | View online |

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