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Notes of Mathematical Method

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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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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
ARW Exercise 1.1 Download PDF (470.64 KiB, 29538 downloads) View online
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ARW Exercise 1.3 Download PDF (1.64 MiB, 14685 downloads) View online
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  • 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: NEW
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ARW Exercise 2.3 Download PDF (657.81 KiB, 3871 downloads) View online
Notes by Shariq Mehtab Syed:
ARW Exercise 2.1 Download PDF (774.94 KiB, 20763 downloads) View online
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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
ARW Articles of Ex 3.1 Download PDF (327.93 KiB, 14300 downloads) View online
ARW Exercise 3.1 Download PDF (630.02 KiB, 28606 downloads) View online
ARW Exercise 3.2 Download PDF (781.02 KiB, 31860 downloads) View online
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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
ARW Exercise 4 Download PDF) (1 MiB, 29212 downloads) View online
  • 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
ARW Exercise 5.1 Download PDF (1.27 MiB, 17472 downloads) View online
ARW Exercise 5.2 Download PDF (395.67 KiB, 27887 downloads) View online

Vetor Space

  • Subspaces
  • Linear combinations and spanning sets
  • Linear dependence and basis
  • Row and column spaces
  • Rand-Alternative method
  • Linear transformations
  • Matrix of linear transformation
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ARW Exercise 6.2 Download PDF (1.23 MiB, 253 downloads) View online
ARW Exercise 6.3 Download PDF (861.66 KiB, 25911 downloads) View online
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
ARW Articles of Exercise 7.1 Download PDF (510.1 KiB, 2511 downloads) View online
ARW Solutions of Exercise 7.1 Download PDF (363.36 KiB, 2792 downloads) View online
ARW Articles of Exercise 7.2 Download PDF (131.18 KiB, 1190 downloads) View online
ARW Solutions of Exercise 7.2 Download PDF (111.31 KiB, 1310 downloads) View online
ARW Articles of Exercise 7.3 Download PDF (157.9 KiB, 1261 downloads) View online
ARW Solutions of Exercise 7.3 Download PDF (519.81 KiB, 3598 downloads) View online
Geometric 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
ARW Exercise 8.1 Download PDF (494.55 KiB, 7230 downloads) View online
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ARW Exercise 8.5 Download PDF (1.46 MiB, 10590 downloads) View online

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