# Basic linear algebra

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This module will give a very brief tutorial on some of the basic terms and ideas of linear algebra. These will include linear independence, span, and basis.

This brief tutorial on some key terms in linear algebra is not meant to replace or be very helpful to those of you trying togain a deep insight into linear algebra. Rather, this brief introduction to some of the terms and ideas of linear algebra ismeant to provide a little background to those trying to get a better understanding or learn about eigenvectors andeigenfunctions, which play a big role in deriving a few important ideas on Signals and Systems. The goal of theseconcepts will be to provide a background for signal decomposition and to lead up to the derivation of the Fourier Series .

## Linear independence

A set of vectors $\forall x, {x}_{i}\in \mathbb{C}^{n}\colon \{{x}_{1}, {x}_{2}, \dots , {x}_{k}\}()$ are linearly independent if none of them can be written as a linear combination of the others.

Linearly Independent
For a given set of vectors, $\{{x}_{1}, {x}_{2}, \dots , {x}_{n}\}()$ , they are linearly independent if ${c}_{1}{x}_{1}+{c}_{2}{x}_{2}+\dots +{c}_{n}{x}_{n}=0$ only when ${c}_{1}={c}_{2}=\dots ={c}_{n}=0$

We are given the following two vectors: ${x}_{1}=\left(\begin{array}{c}3\\ 2\end{array}\right)()$ ${x}_{2}=\left(\begin{array}{c}1\\ 2\end{array}\right)()$ These are linearly independent since ${c}_{1}{x}_{1}=-({c}_{2}{x}_{2})$ only if ${c}_{1}={c}_{2}=0$ . Based on the definition, this proof shows that these vectors are indeed linearly independent. Again, wecould also graph these two vectors (see [link] ) to check for linear independence.

Are $\{{x}_{1}, {x}_{2}, {x}_{3}\}()$ linearly independent? ${x}_{1}=\left(\begin{array}{c}3\\ 2\end{array}\right)()$ ${x}_{2}=\left(\begin{array}{c}1\\ 2\end{array}\right)()$ ${x}_{3}=\left(\begin{array}{c}-1\\ 0\end{array}\right)()$

By playing around with the vectors and doing a little trial and error, we will discover the followingrelationship: ${x}_{1}-{x}_{2}+2{x}_{3}=0$ Thus we have found a linear combination of these threevectors that equals zero without setting the coefficients equal to zero. Therefore, these vectors are not linearly independent !

As we have seen in the two above examples, often times the independence of vectors can be easily seen through a graph.However this may not be as easy when we are given three or more vectors. Can you easily tell whether or not thesevectors are independent from [link] . Probably not, which is why the method used in the above solution becomesimportant. Plot of the three vectors. Can be shown that a linear combination exists among the three, and therefore they are not linear independent.

A set of $m$ vectors in $\mathbb{C}^{n}$ cannot be linearly independent if $m> n$ .

## Span

Span
The span of a set of vectors $\{{x}_{1}, {x}_{2}, \dots , {x}_{k}\}()$ is the set of vectors that can be written as a linear combination of $\{{x}_{1}, {x}_{2}, \dots , {x}_{k}\}()$ $\mathrm{span}(\{{x}_{1}, \dots , {x}_{k}\}())=\{\forall \alpha , {\alpha }_{i}\in \mathbb{C}^{n}\colon {\alpha }_{1}{x}_{1}+{\alpha }_{2}{x}_{2}+\dots +{\alpha }_{k}{x}_{k}\}()$

## Basis

Basis
A basis for $\mathbb{C}^{n}$ is a set of vectors that: (1) spans $\mathbb{C}^{n}$ and (2) is linearly independent.
Clearly, any set of $n$ linearly independent vectors is a basis for $\mathbb{C}^{n}$ .

We are given the following vector ${e}_{i}=\left(\begin{array}{c}0\\ ⋮\\ 0\\ 1\\ 0\\ ⋮\\ 0\end{array}\right)()$ where the $1$ is always in the $i$ th place and the remaining values are zero. Then the basis for $\mathbb{C}^{n}$ is $\{\forall i, i=()\}()$

1 2 n
e i
$\{\forall i, i=()\}()$
1 2 n
e i is called the standard basis .

${h}_{1}=\left(\begin{array}{c}1\\ 1\end{array}\right)()$ ${h}_{2}=\left(\begin{array}{c}1\\ -1\end{array}\right)()$ $\{{h}_{1}, {h}_{2}\}()$ is a basis for $\mathbb{C}^{2}$ .

If $\{{b}_{1}, \dots , {b}_{2}\}()$ is a basis for $\mathbb{C}^{n}$ , then we can express any $x\in \mathbb{C}^{n}$ as a linear combination of the ${b}_{i}$ 's: $\forall \alpha , {\alpha }_{i}\in \mathbb{C}\colon x={\alpha }_{1}{b}_{1}+{\alpha }_{2}{b}_{2}+\dots +{\alpha }_{n}{b}_{n}$

Given the following vector, $x=\left(\begin{array}{c}1\\ 2\end{array}\right)()$ writing $x$ in terms of $\{{e}_{1}, {e}_{2}\}()$ gives us $x={e}_{1}+2{e}_{2}$

Try and write $x$ in terms of $\{{h}_{1}, {h}_{2}\}()$ (defined in the previous example).

$x=\frac{3}{2}{h}_{1}()+\frac{-1}{2}{h}_{2}$

In the two basis examples above, $x$ is the same vector in both cases, but we can express it in many different ways (we give only two out of many, manypossibilities). You can take this even further by extending this idea of a basis to function spaces .

As mentioned in the introduction, these concepts of linear algebra will help prepare you to understand the Fourier Series , which tells us that we can express periodic functions, $f(t)$ , in terms of their basis functions, $e^{i{\omega }_{0}nt}$ .

where we get a research paper on Nano chemistry....?
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
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
nanocopper obvius
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
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 ?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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