# 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.

The notes given here are provided by awesome peoples, who dare to help others. Some of the notes are send by the authors of these notes and other are send by people who didn't write but share these notes as Open Educational Resources (OER). We are thankful to Atiq ur Rehman, Muhammad Farooq, Umer Asghar, Muhammad Tahir Aziz and all those who don't send but write these notes.

We always try our best to add new solutions of more chapters as we are able to manage. If you have notes which you thing are worth to share with other then please contact us from here or email at Admin@MathCity.org.

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

1. $z_1+z_2= (x_1+x_2, y_1+y_2)$
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
• 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

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

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
• 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

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

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

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