Lec01 Introduction to Algebraic Structures Rings and Fields

so welcome to this ah

course on linear algebra

first I want to give

a little introduction

ah then we will be dealing with ah vector spaces and linear maps among them ah as you are aware

the origin of the concept vectors is most notably

the origin of the concept vectors is most n

otably in physics and clearly described and

developed in the geometry of real three spa

ce which I keep calling the universe that we

is an element of a vector space in this sense

the concept of vectors and vector spaces plays its fundamental role in all branches of mathematics

science as well as engineering we will start this course with the abstraction

point and conveniently pass on to geometric

thinking more over it's a geometric perception

thinking more about it, it's a geometric perception

are also beneficial within the framework of the abstract

the theoretical foundations of geometry are most efficiently

formulated in terms of the algebraic structure

suitable pictures and the source of inspiration

for general results ah with this ah in the

first half today I would like to introduce a

abstract vector spaces but before that I would

like to point out ah what prerequisites that

ah one will lead for this course and what

This will lead to this course and what

exactly we will study in the whole course

So first, I will assume that

all of you are familiar with set theory, especially

concepts of sets, maps, and various operations

on the sets and subsets. I also will assume that you are familiar with number systems

such as natural numbers, integers, rational numbers

real numbers, complex numbers, and the standard

operations of addition and multiplication on them. This system I will keep denoting

by N+. This will be for the set of

natural numbers, and I will also remind you

that in the natural numbers, I would like to add zero as a natural number in general terms

for some reference if it is not added to us

but I will insist that zero is a natural

number. Integers I will denote by Z+

they are integers when rational numbers

Q+ or real numbers R

q plus dot or real numbers r

and c plus dot these are the

real numbers and complex number

They are the usual addition and

multiplication on these sets

d multiplication on these sets

and I will assume that all of you are familiar with these number systems and their basic properties

in any case if some properties are needed specifically I will mention it at the time

when we need it and I will also assume that

you are familiar with the basic concepts of

the calculus with t

this is how I would like

to start a linear

algebra and say

that we are going

to study study

of vector spaces and the algebra of

linear operators on vector space

linear operator on vector space

ces usually one studies on vectors