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Mathematical Method Notes Chapter 06 – Vector Spaces

Vector Spaces notes of Mathematical Method chapter 6 for BSc/ADS students preparing for the end-of-term test. These observations are the result of extensive experimentation. In order for all students to readily prepare, we have tried to make these notes as simple as possible. For all officially accredited university students, vector space notes are beneficial. The crew at Study Medium is working hard to provide top-notch notes on mathematical method. if you have any questions comment below.

A collection of vectors that have been added collectively and multiplied (“scaled”) by numbers known as scalars is referred to as a vector space or a linear space. Ordinarily, scalars are regarded as real numbers. However, there aren’t many instances of scalar multiplication with vector spaces when using rational numbers, complex numbers, etc.

Contents

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

Mathematical Method Notes: Chapter 5 – Determinants

Mathematical Method chapter No.5 – Determinants PDF Notes to our BSc/ADS students. So, that’s why we are provide you complete solutions of maths problem, these Determinant solution are absolutely crucial for your university final exams. To ensure that our students can comprehend everything without any trouble, all queries are answered simply. In the end, Study Medium team members is available for our repeated students those contacted through our contact us page.

A determinant is a factor or cause that brings about an event or directly influences a choice. The Latin word for “determining” serves as the word’s primary etymological foundation. It refers to deciding or determining anything as a noun or adjective.

Course Contents

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

Chapter No.3 – System of Linear Equations

Mathematical Method chapter 4 notes – System of Linear Equations Notes S.M. Yusuf, M. Amin, and A. Majeed are the authors of these notes. These notes on mathematical methods have been composed by qualified teachers following a thorough investigation. In case you come into any fallacies while reading Study Medium, notes the team is on call and ready to help.

The main objective of System of Linear Equations eliminating one variable at a time to achieve back-substitution is to solve systems of equations in three variables, also referred to as three-by-three systems. An ordered triple is a solution to a system of three equations in three variables (x, y, and z).

Course Contents

• Preliminaries
• Equivalent equations
• Gaussian elimination method
• Gauss-Jordan elimination method
• Consistency criterion

Mathematical Method Notes: Chapter 3 – Matrices

Mathematical Method notes of chapter 3 – Matrices for our BSc/ADS students on the Mathematical Method by S.M. Yousuf textbook. The author has attempted to discuss every subject in chapter no.3 in these notes. These notes are impressive for those students whose math skills are weak. You will achieve respectable grades within your batch after practicing these questions & answers. Using the comment area, please let us know if you see any errors.

What is Matrix? A matrix is an array of numbers that have been set up in rows and columns to make a rectangular shape. The elements, or entries, of the matrix, are the integers. Numerous branches of mathematics as well as the fields of engineering, physics, economics, and statistics use matrices extensively.

Course Contents

• Introduction
• Algebra of matrices
• Partitioning of matrices
• The inverse of a matrix
• Elementary row operations
• Elementary column operations

Mathematical Method Notes: Chapter 02 – Groups

If your finding Mathematical Method notes for chapter 2 – Groups for help you prepare for exams? if am right, then your in the right place to examine and download the necessary mathematical methods notes. All these method notes of BSc/ADS, our teacher wrote these notes so that they can study them for the end-of-term exam. You can obtain the necessary material in PDF files on the online learning platform known as Study Medium. Additionally, please get in touch with us. Your remarks and total citations will be published.

A set and a binary operation come together to form a group. As an illustration, the collection of integers with an addition operation forms a group, as does the set of real numbers with a binary addition process.

Course Contents

• Definition (axioms of the 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

Mathematical Method Chapter No.1: Complex Numbers

Mathematical method notes pdf for chapter 1 with complete exercise-wise solutions. These pdf notes are specially for all students who are BSc/ADS students. Prepare yourself for university final exams, one by one, for better results. After completing their preparation If you have any questions, please get in touch with us.

These method notes are written by our experts, who are studying at different universities. In this section, we are with complete math preparation for university final mid-term exams as well as final years of exams. By using our mathematical method solution you will be able to solve method paper easily.

What is a Complex Number?

Complex numbers are those that are expressed as a+ib, where a and b are actual numbers and “I” is a fictitious number called an “iota.” I have the value (-1). As an illustration, the complex number 2+3i is made up of the real number 2 (Re) and the imaginary number 3i (IMG).

Course Contents

• Complex number
• Properties of complex numbers
• Argand’s diagram
• De Moivre’s theorem
• Roots of the complex number
• Basic elementary functions
• Logarithmic functions
• Inverse hyperbolic functions
• Inverse trigonometric functions
• Complex power
• Summation of series

Mathematical Method Notes By Prof Naeem-Ul-Haq

Discrete mathematics (MCQs) multiple choice questions in pdf format for exam practice. For students who plan to take the PPSC, FPSC, and NTS exams. These discrete mathematics mcq multiple choice questions are crucial for those. You must complete these MCQs without looking at the correct answers if you want to pass your competitive exam. If you discover any errors, kindly inform us. The study Medium team is available to correct the mistake.

In areas of computer science like computer algorithms, programming languages, cryptography, automated theorem proving, and software development, concepts and notations from discrete mathematics are helpful for analyzing and describing objects and issues.

Chapter 11, The Laplace Transform PDF Notes

Students in the BSc/ADS program are waiting the pdf solutions for Laplace Transformation Mathematical Method Chapter 11. There is a complete set of pdf notes of Mathematical Methods by S.M. Yusuf, A. Majeed, and M. Amin has been released. If you want to write these notes down and prepare the questions and answers later, you can download them in pdf format. You can get full chapter notes on mathematical techniques and calculus from the online education platform Study Medium.

In order to make an equation easier to manage, transformations are used in mathematics to change a variable’s shape. Similar operations are performed via Laplace transformations. In comparison to simply solving differential equations, they convert higher-order differential equations into a polynomial form.

Course Contents

• The Laplace transform
• Properties of the Laplace transform
• Inverse Laplace transform
• Convolution
• Solution of initial value problem

Chapter 10, Higher Order Linear Differential Equations

Higher Order Linear Differential Equations PDF Notes of Chapter 10 of the book on Mathematical Method. These chapter notes were created by qualified professors S.M. Yusuf, A. Majeed, and M. Amin in accordance with the format required by all universities. Mathematical technique notes are created following extensive research. These can be used by BSC/ADS students to study for both their midterm and final exams.

If the related homogeneous equation’s coefficients are all constant functions, then the differential equation has constant coefficients. We can utilize substitution to transform the supplied problem into a quadratic equation, then solve the quadratic equation to get the solutions to the original equation in order to solve higher degree equations.

Course Contents

• Higher order linear differential equations
• Exact equations

Chapter 09, First Order Differential Equations Notes

If you’re looking for a Mathematical Method for First-Order Differential Equations, you might be a BSc/ADS student. Then, you’ve arrived at a page that is upright, where you can download or view your required notes. Famous author S.M. Yusuf spent a lot of time researching before writing and composing these notes. These notes should be thoroughly & meticulously prepared if you are truthful with yourself and wish to get fascinating grades. If any student has questions, they can contact our team; we’re happy to assist. Math notes and multiple choice questions can be found on the well-known website Study Medium.

Notes By Prof Naeem-Ul-Haq

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