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
$x+y=y+x$ for each
$x$ and
$y$ in
$V$
$x+(y+z)()=(x+y)()+z$ for each
$x$ ,
$y$ , and
$z$ in
$V$
There is a unique "zero vector" such that
$x+0=x$ for each
$x$ in
$V$ (0 is the field additive identity)
For each
$x$ in
$V$ there is a unique vector
$-x$ such that
$x+-x=0$
$1x=x$ (1 is the field multiplicative identity)
$({c}_{1}{c}_{2})x={c}_{1}({c}_{2}x)$ for each
$x$ in
$V$ and
${c}_{1}$ and
${c}_{2}$ in
$\u2102$
$c(x+y)=cx+cy$ for each
$x$ and
$y$ in
$V$ and
$c$ in
$\u2102$
$({c}_{1}+{c}_{2})x={c}_{1}x+{c}_{2}x$ for each
$x$ in
$V$ and
${c}_{1}$ and
${c}_{2}$ in
$\u2102$
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?