MTH501 Midterm Short Notes Lectures 1-18

MTH501 Short Notes Mid term · Lec 1 to18 · Website Link:

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Lecture 1 Introduction and Overview

What is Algebra?

Algebra is named in honor of Mohammed Ibn-eMusa al-Khowârizmî. Around 825, he wrote a book entitled Hisb al-jabr u'l muqubalah, , Al-jabr, presented rules for solving equations.

Algebraic Term

The basic unit of an algebraic expression is a term. In general, a term is either a product of a number and with one or more variables.

Algebraic Expressions

An expression is a collection of numbers, variables, and +ve sign or –ve sign, of operations that must make mathematical and logical behaviour.

For example

8X2 +9X −1 is an algebraic expression.

What is Linear Algebra?

One of the most important problems in mathematics is that of solving systems of linear equations. Linear algebra is a branch of mathematics that deals with linear equations and their representations in the vector space using matrices. In other words, linear algebra is the study of linear functions and vectors. It is one of the most central topics of mathematics. Most modern geometrical concepts are based on linear algebra.

Elementary Linear Algebra

Elementary linear algebra introduces students to the basics of linear algebra. This includes simple matrix operations, various computations that can be done on a system of linear equations, and certain aspects of vectors. Some important terms associated with elementary linear algebra are given below:

Scalars

A scalar is a quantity that only has magnitude and not direction. It is an element that is used to define a vector space. In linear algebra, scalars are usually real numbers.

Vectors

A vector is an element in a vector space. It is a quantity that can describe both the direction and magnitude of an element.

Vector Space

The vector space consists of vectors that may be added together and multiplied by scalars.

Matrix

A matrix is a rectangular array wherein the information is organized in the form of rows and columns. Most linear algebra properties can be expressed in terms of a matrix.

Matrix Operations

These are simple arithmetic operations such as addition, subtraction, and multiplication that can be conducted on matrices.

Applications of Linear algebra

Linear algebra makes it possible to work with large arrays of data. It has many applications in many diverse fields, such as

•  Computer Graphics,
•  Electronics,
•  Chemistry,
•  Biology,
•  Differential Equations,
•  Economics,
•  Business,
•  Psychology,
•  Engineering,
•  Analytic Geometry,
•  Chaos Theory,
•  Cryptography,
•  Fractal Geometry,
•  Game Theory,
•  Graph Theory,
•  Linear Programming,
•  Operations Research

Why using Linear Algebra?

Linear Algebra is then useful for solving problems in such applications in topics such as Physics, Fluid Dynamics, Signal Processing and, more generally Numerical Analysis.