# Review of linear algebra

 Page 2 / 2

## Dimension

Let $V$ be a vector space with basis $B$ . The dimension of $V$ , denoted $\mathrm{dim}(V)$ , is the cardinality of $B$ .

Every vector space has a basis.

Every basis for a vector space has the same cardinality.

$\implies \mathrm{dim}(V)$ is well-defined .

If $\mathrm{dim}(V)$ , we say $V$ is finite dimensional .

## Examples

vector space field of scalars dimension
$\mathbb{R}^{N}$ $\mathbb{R}$
$\mathbb{C}^{N}$ $\mathbb{C}$
$\mathbb{C}^{N}$ $\mathbb{R}$

Every subspace is a vector space, and therefore has its own dimension.

Suppose $(S=\{{u}_{1}, , {u}_{k}\})\subseteq V$ is a linearly independent set. Then $\mathrm{dim}()=$

## Facts

• If $S$ is a subspace of $V$ , then $\mathrm{dim}(S)\le \mathrm{dim}(V)$ .
• If $\mathrm{dim}(S)=\mathrm{dim}(V)$ , then $S=V$ .

## Direct sums

Let $V$ be a vector space, and let $S\subseteq V$ and $T\subseteq V$ be subspaces.

We say $V$ is the direct sum of $S$ and $T$ , written $V=(S, T)$ , if and only if for every $v\in V$ , there exist unique $s\in S$ and $t\in T$ such that $v=s+t$ .

If $V=(S, T)$ , then $T$ is called a complement of $S$ .

$V={C}^{}=\left\{f:\mathbb{R}\mathbb{R}|f\text{is continuous}\right\}$ $S=\text{even funcitons in}{C}^{}$ $T=\text{odd funcitons in}{C}^{}$ $f(t)=\frac{1}{2}(f(t)+f(-t))+\frac{1}{2}(f(t)-f(-t))$ If $f=g+h={g}^{}+{h}^{}$ , $g\in S$ and ${g}^{}\in S$ , $h\in T$ and ${h}^{}\in T$ , then $g-{g}^{}={h}^{}-h$ is odd and even, which implies $g={g}^{}$ and $h={h}^{}$ .

## Facts

• Every subspace has a complement
• $V=(S, T)$ if and only if
• $S\cap T=\{0\}$
• $=V$
• If $V=(S, T)$ , and $\mathrm{dim}(V)$ , then $\mathrm{dim}(V)=\mathrm{dim}(S)+\mathrm{dim}(T)$

Invoke a basis.

## Norms

Let $V$ be a vector space over $F$ . A norm is a mapping $(V, F)$ , denoted by $()$ , such that forall $u\in V$ , $v\in V$ , and $\in F$

• $(u)> 0$ if $u\neq 0$
• $(u)=\left|\right|(u)$
• $(u+v)\le (u)+(v)$

## Examples

Euclidean norms:

$x\in \mathbb{R}^{N}$ : $(x)=\sum_{i=1}^{N} {x}_{i}^{2}^{\left(\frac{1}{2}\right)}$ $x\in \mathbb{C}^{N}$ : $(x)=\sum_{i=1}^{N} \left|{x}_{i}\right|^{2}^{\left(\frac{1}{2}\right)}$

## Induced metric

Every norm induces a metric on $V$ $d(u, v)\equiv (u-v)$ which leads to a notion of "distance" between vectors.

## Inner products

Let $V$ be a vector space over $F$ , $F=\mathbb{R}$ or $\mathbb{C}$ . An inner product is a mapping $V\times VF$ , denoted $\dot$ , such that

• $v\dot v\ge 0$ , and $(v\dot v=0, v=0)$
• $u\dot v=\overline{v\dot u}$
• $au+bv\dot w=a(u\dot w)+b(v\dot w)$

## Examples

$\mathbb{R}^{N}$ over: $x\dot y=x^Ty=\sum_{i=1}^{N} {x}_{i}{y}_{i}$

$\mathbb{C}^{N}$ over: $x\dot y=(x)y=\sum_{i=1}^{N} \overline{{x}_{i}}{y}_{i}$

If $(x=\left(\begin{array}{c}{x}_{1}\\ \\ {x}_{N}\end{array}\right))\in \mathbb{C}$ , then $(x)\equiv \left(\begin{array}{c}\overline{{x}_{1}}\\ \\ \overline{{x}_{N}}\end{array}\right)^T$ is called the "Hermitian," or "conjugatetranspose" of $x$ .

## Triangle inequality

If we define $(u)=u\dot u$ , then $(u+v)\le (u)+(v)$ Hence, every inner product induces a norm.

## Cauchy-schwarz inequality

For all $u\in V$ , $v\in V$ , $\left|u\dot v\right|\le (u)(v)$ In inner product spaces, we have a notion of the angle between two vectors: $((u, v)=\arccos \left(\frac{u\dot v}{(u)(v)}\right))\in \left[0 , 2\pi \right)$

## Orthogonality

$u$ and $v$ are orthogonal if $u\dot v=0$ Notation: $(u, v)$ .

If in addition $(u)=(v)=1$ , we say $u$ and $v$ are orthonormal .

In an orthogonal (orthonormal) set , each pair of vectors is orthogonal (orthonormal).

## Orthonormal bases

An Orthonormal basis is a basis $\{{v}_{i}\}$ such that ${v}_{i}\dot {v}_{i}={}_{ij}=\begin{cases}1 & \text{if i=j}\\ 0 & \text{if i\neq j}\end{cases}$

The standard basis for $\mathbb{R}^{N}$ or $\mathbb{C}^{N}$

The normalized DFT basis ${u}_{k}=\frac{1}{\sqrt{N}}\left(\begin{array}{c}1\\ e^{-(i\times 2\pi \frac{k}{N})}\\ \\ e^{-(i\times 2\pi \frac{k}{N}(N-1))}\end{array}\right)$

## Expansion coefficients

If the representation of $v$ with respect to $\{{v}_{i}\}$ is $v=\sum {a}_{i}{v}_{i}$ then ${a}_{i}={v}_{i}\dot v$

## Gram-schmidt

Every inner product space has an orthonormal basis. Any (countable) basis can be made orthogonal by theGram-Schmidt orthogonalization process.

## Orthogonal compliments

Let $S\subseteq V$ be a subspace. The orthogonal compliment $S$ is ${S}^{}=\{u\colon u\in V\land (u\dot v=0)\land \forall v\colon v\in S\}$ ${S}^{}$ is easily seen to be a subspace.

If $\mathrm{dim}(v)$ , then $V=(S, {S}^{})$ .

If $\mathrm{dim}(v)$ , then in order to have $V=(S, {S}^{})$ we require $V$ to be a Hilbert Space .

## Linear transformations

Loosely speaking, a linear transformation is a mapping from one vector space to another that preserves vector space operations.

More precisely, let $V$ , $W$ be vector spaces over the same field $F$ . A linear transformation is a mapping $T:VW$ such that $T(au+bv)=aT(u)+bT(v)$ for all $a\in F$ , $b\in F$ and $u\in V$ , $v\in V$ .

In this class we will be concerned with linear transformations between (real or complex) Euclidean spaces , or subspaces thereof.

## Image

$()$ T w w W T v w for some v

## Nullspace

Also known as the kernel: $\mathrm{ker}(T)=\{v\colon v\in V\land (T(v)=0)\}$

Both the image and the nullspace are easily seen to be subspaces.

## Rank

$\mathrm{rank}(T)=\mathrm{dim}(())$ T

## Nullity

$\mathrm{null}(T)=\mathrm{dim}(\mathrm{ker}(T))$

## Rank plus nullity theorem

$\mathrm{rank}(T)+\mathrm{null}(T)=\mathrm{dim}(V)$

## Matrices

Every linear transformation $T$ has a matrix representation . If $T:{𝔼}^{N}{𝔼}^{M}$ , $𝔼=\mathbb{R}$ or $\mathbb{C}$ , then $T$ is represented by an $M\times N$ matrix $A=\begin{pmatrix}{a}_{11} & & {a}_{1N}\\ & & \\ {a}_{M1} & & {a}_{MN}\\ \end{pmatrix}$ where $\left(\begin{array}{c}{a}_{1i}\\ \\ {a}_{Mi}\end{array}\right)=T({e}_{i})$ and ${e}_{i}=\left(\begin{array}{c}0\\ \\ 1\\ \\ 0\end{array}\right)$ is the $i^{\mathrm{th}}$ standard basis vector.

A linear transformation can be represented with respect to any bases of $𝔼^{N}$ and $𝔼^{M}$ , leading to a different $A$ . We will always represent a linear transformation using the standard bases.

## Column span

$\mathrm{colspan}(A)==()$ A

## Duality

If $A:{\mathbb{R}}^{N}{\mathbb{R}}^{M}$ , then $\mathrm{ker}(A)^{}=()$ A

If $A:{\mathbb{C}}^{N}{\mathbb{C}}^{M}$ , then $\mathrm{ker}(A)^{}=()$ A

## Inverses

The linear transformation/matrix $A$ is invertible if and only if there exists a matrix $B$ such that $AB=BA=I$ (identity).

Only square matrices can be invertible.

Let $A:{𝔽}^{N}{𝔽}^{N}$ be linear, $𝔽=\mathbb{R}$ or $\mathbb{C}$ . The following are equivalent:

• $A$ is invertible (nonsingular)
• $\mathrm{rank}(A)=N$
• $\mathrm{null}(A)=0$
• $\det A\neq 0$
• The columns of $A$ form a basis.

If $A^{(-1)}=A^T$ (or $(A)$ in the complex case), we say $A$ is orthogonal (or unitary ).

#### Questions & Answers

how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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
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
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!