Review of linear algebra

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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

are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
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
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
nanocopper obvius
Alexandre
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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Source:  OpenStax, Statistical signal processing. OpenStax CNX. Jun 14, 2004 Download for free at http://cnx.org/content/col10232/1.1
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