# 15.1 Vector spaces

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This module will define what a vector space is and provide useful examples to the reader.

## Introduction

Vector space
A vector space $S$ is a collection of "vectors" such that (1) if ${f}_{1}\in S\implies \alpha {f}_{1}\in S$ for all scalars $\alpha$ (where $\alpha \in \mathbb{R}$ , $\alpha \in \mathbb{C}$ , or some other field) and (2) if ${f}_{1}\in S$ , ${f}_{2}\in S$ , then $({f}_{1}+{f}_{2})\in S$
To define an vector space, we need
• A set of things called "vectors" ( $X$ )
• A set of things called "scalars" that form a field ( $A$ )
• A vector addition operation (  )
• A scalar multiplication operation ( $*$ )
The operations need to have all the properties of givenbelow. Closure is usually the most important to show.

## Vector spaces

If the scalars $\alpha$ are real, $S$ is called a real vector space .

If the scalars $\alpha$ are complex, $S$ is called a complex vector space .

If the "vectors" in $S$ are functions of a continuous variable, we sometimes call $S$ a linear function space

## Properties

We define a set $V$ to be a vector space if

1. $x+y=y+x$ for each $x$ and $y$ in $V$
2. $x+(y+z)()=(x+y)()+z$ for each $x$ , $y$ , and $z$ in $V$
3. There is a unique "zero vector" such that $x+0=x$ for each $x$ in $V$ (0 is the field additive identity)
4. For each $x$ in $V$ there is a unique vector $-x$ such that $x+-x=0$
5. $1x=x$ (1 is the field multiplicative identity)
6. $\left({c}_{1}{c}_{2}\right)x={c}_{1}\left({c}_{2}x\right)$ for each $x$ in $V$ and ${c}_{1}$ and ${c}_{2}$ in $ℂ$
7. $c(x+y)=cx+cy$ for each $x$ and $y$ in $V$ and $c$ in $ℂ$
8. $({c}_{1}+{c}_{2})x={c}_{1}x+{c}_{2}x$ for each $x$ in $V$ and ${c}_{1}$ and ${c}_{2}$ in $ℂ$

## Examples

• $\mathbb{R}^{n}=\mathrm{real vector space}$
• $\mathbb{C}^{n}=\mathrm{complex vector space}$
• ${L}^{1}(\mathbb{R})=\{f(t)\colon \int_{()} \,d t\}$ f t f t is a vector space
• is a vector space
• ${L}^{2}(\mathbb{R})=\{f(t)\colon \int_{()} \,d t\}$ f t 2 f t finite energy signals is a vector space
• ${\ell }^{1}(\mathbb{Z})$ , ${\ell }^{2}(\mathbb{Z})$ , ${\ell }^{\infty }(\mathbb{Z})$ are vector spaces
• The collection of functions piecewise constant between the integers is a vector space

• ${ℝ}_{+}^{2}=\{\left(\begin{array}{c}{x}_{0}\\ {x}_{1}\end{array}\right)\colon ({x}_{0}> 0)\land ({x}_{1}> 0)\}$ is not a vector space. $\left(\begin{array}{c}1\\ 1\end{array}\right)\in {ℝ}_{+}^{2}$ , but $\forall \alpha , \alpha < 0\colon \alpha \left(\begin{array}{c}1\\ 1\end{array}\right)\notin {ℝ}_{+}^{2}$
• $D=\{\forall z, \left|z\right|\le 1\colon z\in \mathbb{C}\}$ is not a vector space. $({z}_{1}=1)\in D$ , $({z}_{2}=i)\in D$ , but $({z}_{1}+{z}_{2})\notin D$ , $\left|{z}_{1}+{z}_{2}\right|=\sqrt{2}> 1$

Vector spaces can be collections of functions, collections of sequences, as well as collections of traditionalvectors ( i.e. finite lists of numbers)

#### Questions & Answers

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
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