Review of linear algebra

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Vector spaces are the principal object of study in linear algebra. A vector space is always defined with respectto a field of scalars.

Fields

A field is a set $F$ equipped with two operations, addition and mulitplication, and containing two special members 0 and 1( $0\neq 1$ ), such that for all $\{a, b, c\}\in F$

• $(a+b)\in F$
• $a+b=b+a$
• $\left(a+b\right)+c=a+\left(b+c\right)$
• $a+0=a$
• there exists $-a$ such that $a+-a=0$
• $ab\in F$
• $ab=ba$
• $abc=abc$
• $a1=a$
• there exists $a^{(-1)}$ such that $aa^{(-1)}=1$
• $a(b+c)=ab+ac$
More concisely
• $F$ is an abelian group under addition
• $F$ is an abelian group under multiplication
• multiplication distributes over addition

,,

Vector spaces

Let $F$ be a field, and $V$ a set. We say $V$ is a vector space over $F$ if there exist two operations, defined for all $a\in F$ , $u\in V$ and $v\in V$ :

• vector addition: ( $u$ , $v$ ) $(u+v)\in V$
• scalar multiplication: ( $a$ , $v$ ) $av\in V$
and if there exists an element denoted $0\in V$ , such that the following hold for all $a\in F$ , $b\in F$ , and $u\in V$ , $v\in V$ , and $w\in V$
• $u+\left(v+w\right)=\left(u+v\right)+w$
• $u+v=v+u$
• $u+0=u$
• there exists $-u$ such that $u+-u=0$
• $a(u+v)=au+av$
• $(a+b)u=au+bu$
• $abu=abu$
• $1u=u$
More concisely,
• $V$ is an abelian group under plus
• Natural properties of scalar multiplication

Examples

• $\mathbb{R}^{N}$ is a vector space over
• $\mathbb{C}^{N}$ is a vector space over
• $\mathbb{C}^{N}$ is a vector space over
• $\mathbb{R}^{N}$ is not a vector space over
The elements of $V$ are called vectors .

Euclidean space

Throughout this course we will think of a signal as a vector $x=\left(\begin{array}{c}{x}_{1}\\ {x}_{2}\\ \\ {x}_{N}\end{array}\right)=\begin{pmatrix}{x}_{1} & {x}_{2} & & {x}_{N}\\ \end{pmatrix}^T$ The samples $\{{x}_{i}\}$ could be samples from a finite duration, continuous time signal, for example.

A signal will belong to one of two vector spaces:

Real euclidean space

$x\in \mathbb{R}^{N}$ (over)

Complex euclidean space

$x\in \mathbb{C}^{N}$ (over)

Subspaces

Let $V$ be a vector space over $F$ .

A subset $S\subseteq V$ is called a subspace of $V$ if $S$ is a vector space over $F$ in its own right.

$V=\mathbb{R}^{2}$ , $F=\mathbb{R}$ , $S=\text{any line though the origin}$ .

Are there other subspaces?

$S\subseteq V$ is a subspace if and only if for all $a\in F$ and $b\in F$ and for all $s\in S$ and $t\in S$ , $(as+bt)\in S$

Linear independence

Let ${u}_{1},,{u}_{k}\in V$ .

We say that these vectors are linearly dependent if there exist scalars ${a}_{1},,{a}_{k}\in F$ such that

$\sum_{i=1}^{k} {a}_{i}{u}_{i}=0$
and at least one ${a}_{i}\neq 0$ .

If only holds for the case ${a}_{1}=={a}_{k}=0$ , we say that the vectors are linearly independent .

$1\left(\begin{array}{c}1\\ -1\\ 2\end{array}\right)-2\left(\begin{array}{c}-2\\ 3\\ 0\end{array}\right)+1\left(\begin{array}{c}-5\\ 7\\ -2\end{array}\right)=0$ so these vectors are linearly dependent in $\mathbb{R}^{3}$ .

Spanning sets

Consider the subset $S=\{{v}_{1}, {v}_{2}, , {v}_{k}\}$ . Define the span of $S$ $\equiv \mathrm{span}(S)\equiv \{\sum_{i=1}^{k} {a}_{i}{v}_{i}\colon {a}_{i}\in F\}$

Fact: $$ is a subspace of $V$ .

$V=\mathbb{R}^{3}$ , $F=\mathbb{R}$ , $S=\{{v}_{1}, {v}_{2}\}$ , ${v}_{1}=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)$ , ${v}_{2}=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$ $=\text{xy-plane}$ .

Aside

If $S$ is infinite, the notions of linear independence and span are easily generalized:

We say $S$ is linearly independent if, for every finite collection ${u}_{1},,{u}_{k}\in S$ , ( $k$ arbitrary) we have $(\sum_{i=1}^{k} {a}_{i}{u}_{i}=0)\implies \forall i\colon {a}_{i}=0$ The span of $S$ is $=\{\sum_{i=1}^{k} {a}_{i}{u}_{i}\colon {a}_{i}\in F\land {u}_{i}\in S\land (k)\}$

In both definitions, we only consider finite sums.

Bases

A set $B\subseteq V$ is called a basis for $V$ over $F$ if and only if

• $B$ is linearly independent
• $=V$
Bases are of fundamental importance in signal processing. They allow us to decompose a signal into building blocks (basisvectors) that are often more easily understood.

$V$ = (real or complex) Euclidean space, $\mathbb{R}^{N}$ or $\mathbb{C}^{N}$ . $B=\{{e}_{1}, , {e}_{N}\}\equiv \text{standard basis}$ ${e}_{i}=\left(\begin{array}{c}0\\ \\ 1\\ \\ 0\end{array}\right)$ where the 1 is in the $i^{\mathrm{th}}$ position.

$V=\mathbb{C}^{N}$ over. $B=\{{u}_{1}, , {u}_{N}\}$ which is the DFT basis. ${u}_{k}=\left(\begin{array}{c}1\\ e^{-(i\times 2\pi \frac{k}{N})}\\ \\ e^{-(i\times 2\pi \frac{k}{N}(N-1))}\end{array}\right)$ where $i=\sqrt{-1}$ .

Key fact

If $B$ is a basis for $V$ , then every $v\in V$ can be written uniquely (up to order of terms) in the form $v=\sum_{i=1}^{N} {a}_{i}{v}_{i}$ where ${a}_{i}\in F$ and ${v}_{i}\in B$ .

Other facts

• If $S$ is a linearly independent set, then $S$ can be extended to a basis.
• If $=V$ , then $S$ contains a basis.

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
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
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what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
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it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
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Anassong
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there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
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