Bu ders doğrusal cebir ikili dizinin birincisidir. Doğrusal uzaylar kavramı, doğrusal işlemciler, matris gösterimleri ve denklem sistemlerinin hesaplanabilmesi için temel araçlar vb. konuları içermektedir. Ders gerçek yaşamdan gelen uygulamaları da tanıtmaya önem veren “içerikli yaklaşımla” tasarlanmıştır. Bölümler: Bölüm 1: Doğrusal Cebirin Matematikdeki Yeri ve Kapsamı Bölüm 2: Düzlemdeki Vektörlerin Öğrettikleri Bölüm 3: İki Bilinmeyenli Denklemlerin Öğrettikleri Bölüm 4: Doğrusal Uzaylar Bölüm 5: Fonksiyon Uzayları ve Fourier Serileri Bölüm 6: Doğrusal İşlemciler ve Dönüşümler Bölüm 7: Doğrusal İşlemcilerden Matrislere Geçiş Bölüm 8: Matris İşlemleri ----------- This is the first of the sequence of two courses. It develops the fundamental concepts in linear spaces, linear operators, matrix representations and basic tools for calculations with systems of equations. The course is designed with a “content based” emphasis, answering the “why” and “where“ of the topics, as much as the traditional “what” and “how” leading to “definitions” and “proofs”. Chapters: Chapter 1: Place and Contents of Linear Algebra Cebirin Chapter 2: Learning From Vectors in the Plane Chapter 3: Learning From Equations For Two Unknowns Chapter 4: Linear Spaces Chapter 5: Function Spaces and Fourier Series Chapter 6: Linear Operators and Transformations Chapter 7: From Linear Operators to Matrices Chapter 8: Matrix Operations ----------- Kaynak: Attila Aşkar, “Doğrusal cebir”. Bu kitap dört ciltlik dizinin üçüncü cildidir. Dizinin diğer kitapları Cilt 1 “Tek değişkenli fonksiyonlarda türev ve entegral”, Cilt 2: "Çok değişkenli fonksiyonlarda türev ve entegral" ve Cilt 4: “Diferansiyel denklemler” dir. Source: Attila Aşkar, Linear Algebra, Volume 3 of the set of Vol1: Calculus of Single Variable Functions, Volume 2: Calculus of Multivariable Functions and Volume 4: Differential Equations.
Gerçel ve Karmaşık Değerli Fourier Serileri / Real and Complex Valued Fourier Series
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Doğrusal Cebir I: Uzaylar ve İşlemciler / Linear Algebra I: Spaces and Operators
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Fonksiyon Uzayları ve Fourier Serileri / Function Spaces and Fourier Series
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Attila AşkarProf. Dr.
Matematik Bölümü (Department of Mathematics)
Hello.
We discovered an important treasure in our preceding session.
Any range of these sine and cosine functions,
This range wherein sufficient number of pi
Let us take as half, we discover that each other.
This is in the plane or in space, i, j,
Perhaps more important than the k unit vector perpendicular to each other.
They are now using what we call a Fourier Series
cosine and sine function in terms of the opening series.
Here's a not a zero surplus property
this short coming opposed to zero for a zero cosine we say that k is zero,
We saw a bit of that feature alone.
We take him out and suddenly we are launching this series
or we could also start from scratch.
What we do now to calculate the coefficients in this expansion.
We've seen it before, but here is a reminder that you're here, I say.
We encounter this cosine and sine of the square.
I wrote here just cosine, cosine square last summer in terms of double angle
equal time is going and where the cosine
cons plus for the full period a catheter a full range of motion
He's taking it gives integral zero, leaving only the remains integral to this one.
In the meantime, of course, be one of the integral will be a minus.
A split will be minus two times.
I remember this one.
Turning now how do we take this function previous
In three dimensions, as we see in the session v1, v2,
We stood there with a good vector to find v3 coefficient stood with j
k'yl stood out, and then we saw it all with this single base function
hit that takes into account that this hit domestic product integration
Coming to the opposite, we find that the coefficients.
Currently, A0, A1, A2, forever b1, b2,
be forever unknown coefficients that we have infinite knowledge.
We want to solve them.
The greatest support for these functions to be perpendicular to each other here.
Now let's take the integral combined to hit this function.
Here he has hit the merger integration.
Here is the integral of a0, all of them in the cosine and sine integrals
Integral also gives zero.
The other because the other cosine, perpendicular to the sinuses.
this function is still to find a k of the cosine sine
We're getting hit in the integration with other kPa.
Not all other terms, wherein the endless term a0'l terms,
where the infinite sine terms, all these integrals gives zero.
Only here we hit we hit kosinüsl kosinüsl f we have the integration.
All of them fell, leaving only the term acute cholecystitis.
It also comes kosinüsl to hit the cosine square.
E We see here a cosine squared divided by two
b minus a giving, here are the current front.
This means that we are each a k from an infinite variable equations
We found the solving ability.
This work captures a very rich treasure in my head since we
We therefore also be orthogonal.
If not for all the other unknowns infinite integrals
need to solve the equation forever.
This is how it is resolved, it can be solved,
You can imagine would be resolved easily.
Likewise, if we hit the sine integral calculus this series,
One of sine integral gives zero, the product of kosinüsl sine
He gives zero, just stay the same K characteristics in terms of sine sinus sinus here.
It's like a symmetrical equivalent as you can see here
Coming cosine sine instead, we find a k and b k instead,
here comes the square instead of sine cosine square but they were both integral in
We saw the same thing because it gives the sine-squared said, will have a negative kosinüsl of.
As such, these coefficients
We have found that our unknown is this flow, and a0 and see.
On the right side, the left side for a given function,
known cosine function so that when we calculate integrals
This will be a number known.
Similarly, the sine of.
That means that the coefficients are holding as follows.
See a0 has a slight difference.
There 2p because this integral in the denominator 2p wherein b minus a bun.
But here there is a divide b minus two.
which is different from k0 cases
The integral of the same chord with a similar product cosine integral f
a'yl b in between the limits.
Here b k are the sine for.
That coefficients of this series to be situated, f (fx) we get.
I took a matter of convenience here forever so start from scratch.
Because k is equal to zero when will this one so a0'l term.
k is zero when the sinus zero, it will fall.
Therefore, the ability to write so well consisted of little more briefly.
These coefficients are the coefficients of the accounts that we have above.
In many of these applications a and b are symmetric
such income is taken as the range and naturally.
That may be due to the nature of the problems in our choice Or,
it brings.
Minus p, b is the integral of this goes as p
not a fundamental change in the formula.
rather than a minus b, p, b, less than p,
eksi p'den p'ye, eksi p'den p'ye gider.
b instead of p, when we put a pin instead of a minus.
So it's come down a bit more special, but it's one of the above formulas
We will see now is a useful side.
This function p, x instead of two
pi times the same as it occurs with a full count sheep to çarpılmış.
An essential feature of this Fourier series, so it's also FX
x If we increase instead of the up to 2p,
so this periodic function itself is obtained.
So as geometric, plot Or, as we so far from A to B
We want to achieve function, this is our goal.
But what we say is going out of it, we find ötelenmiş of this same function.
Similarly, we take a rather less symmetric interval p
we receive, we find that the middle pin but still function rather than b
If we say that what is happening outside the same this function ötelenmiş
We find a fundamental feature of the Fourier series.
Because the sine and cosine functions are periodic.
The function obtained for functions we use this base
We get periodic ötelenmiş.
It's a natural expectation but there to remind you.
Now sometimes functions are symmetrical,
The minus x fx we say that for symmetric function.
For example, if you take the square of x squared minus x is equal to x squared.
Cosine function is symmetrical again.
If we see the symmetric function in this integration would be facilitated on a like.
This means taking the integral functions of up to minus p p because it is symmetrical,
which means you get twice the integral from zero up to p.
Thus, here the integral from minus p p will take place from zero
We take up to p, but we are getting twice.
This formula is similar as in integral that other coefficients a k
We find the integral of minus multiplied by the cosine for up to p to p.
Again f
for a symmetric function, it is also symmetrical cosine function.
If you change because x minus x minus x cosine of x is equal to cosine
which is symmetrical to the symmetrical, two multiplication it asymmetrical.
Therefore integral to the negative taken up to p than p,
It would be up to scratch of p twice.
The tactics also be something more interesting, but for sinus anti-symmetric symmetrical.
These two product to hit the minus side of the plus
It would have to be given the diminished as anti-symmetric.
Integration with up to zero from minus p
p until the scratch is less marked on it.
This means that each one takes.
Here's our goose is as follows: which both flow in both bk
we only went to the chord sequence in the series.
These two types available; the function symmetric
It may be in the range of -p and p,
or function is given to us in the pan range of 0,
We range from 0 to p less to do symmetric
We extend the range of the symmetrically as artificial, artificial way.
But here's a gain happens because our two
Instead of dealing with different coefficients of a single series
that we are dealing with series of EC sex.
This function than to be symmetrical or not symmetrical
We extend our function as artificial as symmetrical,
but more of them will be placed in our application.
This has equivalent; Our function is anti-symmetric, for example a cube x
If you receive the -x anti-symmetric function of the cube of the cube is equal to minus x,
It provides the same functionality of a sinus missing, it is also anti-symmetric.
Now here is the integral f anti-symmetrical
We thought we encounter a brew with anti-symmetric function,
No part up to each person with p from 0 to 0 it from -p'.
If one side off the other side minus plus interest, plus a side
If it turns out negative on the other side; these results cancel each other.
Destroy the coefficient of a0 is 0.
Similarly, a symmetrical cosine function,
It did this for the anti-symmetric product is already anti-symmetric
It will be a function, here again with the -p
An interval between 0 0 and p
entergral between this range takes another.
If somewhere in the plus minus comes out on the other side, then it flows from the 0's going on,
so are the full equivalent of a previous state, but through the sinuses.
But when we look at the f multiplication by sine for anti-symmetric,
so negative as it occurs,
sinus means that the product will be a minus ten is made symmetrical.
In Symmetrical half of the integral function
which it is equal to from 0 to -p' of from 0 up to p.
Therefore, take this integral p instead of the -p' 0
If we multiply by 2, we take p to achieve the same result.
So we see that in many instances, integration between -5 and 5 x squared
x frame is integrally 2 times between 0 and 5 because it is a symmetric function.
This often encountered something.
So both be found here as well as a series
Located just becomes a series b.
Again, this former function as an anti-symmetric
What does this automatically, but if already received anti-symmetric functions and if the
This function has given us so that we JOIN with 0 per interval substantially symmetric
0 -p range between anti-symmetric artificially extend,
but we do it because we have a gain,
We can only be obtained with a series denominated.
These gains because in practice how fast,
There will be things we can do short of what the results are the same.
If we look at these very complex valued integral
we can not play with much symmetry.
We say here that i e to the virtual
the same unit that is to be the square root of -1
As in the other two sides of the cosine and sine
k p x divide them as pure as it was when you receive
We have seen that the orthogonal functions.
This function takes a, say that they together form a base
Or, a full function other vectors
they constitute a full team
The cluster will create a base on the bottom that create them
in a very different place to prove we are accepting it possible.
Now where do we take this function will multiply the base complex.
Remember if you have a complex inner product base
going to the first complex mappings.
It -p via e X, we will hit with customers.
Üssül because you get a k may be different than k.
See them all here would be that when we hit 0,
except one, when it is equal to k k base.
Then when we received the first of which 1 will be the integration
b b-2P was a through and only those ck
but equal to k and k will survive the base of this collection,
0 will be different, so although still unknown infinite
This indicator k c c c minus minus infinity to plus infinity forever going up,
There are infinite unknown, but it all every one
the unknowns we get from the equation, given on the left side for this particular function,
We can calculate the integral of the left side of the check account.
So we can reduce this one also an unknown ck base
ck get easier writing this here because the denominator 2p
it would have become integral to come.
Now we work with here, but if the exponential function f
If this is the equivalent of real sine and cosine series,
There's nothing you can do now does not function f is a complex function possible,
f you account this factor, knowing well that it is a complex function.
But in many cases is a real function f,
I'm talking to you in a time such as voice and
a function that indicates an oscillation in space.
So this is a real function, but this complex
We have the opportunity to show in terms of functions.
Here we can show as an equivalent,
From this theoretical equivalence of the major issues in terms of maintenance,
How that Euler's formula thus cosine theta plus
e to sine of theta times i was them i teta'y
We can open each e to the k,
also here because it is real fun
i'yi eksiye çevirirsek c'yi de eşle
Taking the complex conjugate which gives f again.
So here's what this means complex conjugate Let c complex conjugate,
where i is to multiply the -i.
This again we get two different impressions given for f and f,
Let them together, and returns again to f half.
This will allow us to CK-ease
Let's write this complex so as ac- ibk
the imaginary part of the real part of this summer could be a plus for the number
but we use such accounts for bringing a little more simplicity in negative letter.
The complex of data that is changing the i c -ie.
Now here I use Euler's formula ii via e theta,
If we say that part of the theta, which is a plus i sine cosine.
There is sour if it is minus cosine sine.
It's there in front of CK, CK-ibk he described the ac- we
He also had before it flows + ibk ck base.
As you can see here, here Cons ck cosine
plus sinus come here chord plus i ib'li of
minus cosine sine coming here because it is e to the minus.
This is a simple algebraic collection, so here is a combination of real and I gather,
If we gather together this virtual account,
See here the size of the flows here we let this one get two terms,
In the beginning there comes a cosine 1 divided by 2.
Here, too, there are cosine flow, beginning a 1 divided by 2.
But when you look at the flow here i'l sinus plus,
When you look at sine minus i got here to i'l of Ak'yl.
So liter, it is taking another,
so you can see all of the individual öbürkü.
Only real and are forced to kosinüslü are only ak'yl
remain as united here is a dilemma I always split a combined offer.
See here plus minus
I think for that you get a hit, it's time see sinuses.
Here is also a plus to minus i give,
see once again going one more sinus.
There is also the dilemma of a divided front of each, have collected by the liter, and destroy each other,
simplifies; öbürkü are also collected, transformed into a complete split of the dilemma.
As you can see, there appeared a Fourier series,
we know Fourier series output.
So this shows the real representation of my equivalence with exponential notation,
it is possible to pass from one to the other.
Now there is a fast Fourier series.
This is actually a very interesting [cough] and is based on a simple observation.
Now we'll accounts this coefficient ck.
But for them it is infinite, is not worth doing them manually,
do the impossible.
Moreover IF YOU that my voice will post a phone elsewhere
This blend function.
Therefore, the only thing that can be done,
the computer account to the numerical integration.
Integral collection.
What are we doing in Integral?
We are in any value, we find the value.
For example, let's worth xn.
Here we take the value of x in xn.
We put in place the delta delta x, we collect them.
When we collect them, see how something interesting occurs.
This here is exponential
deltal of i'l of pI consists term.
See him, he'll throw down there taking the term.
This exponential term, we reserve the kn's see where i p e to the minus delta p.
These are all well known.
E kN say multiply exponentially,
This means taking the number nk'yınc forces.
So, I have this number,
I say here that a beta number in brackets,
this is a complex issue; kN to any çarpılmış of
This means you get the beta kn'yinc forces.
Now this may seem extremely simple; but practically,
When you think in terms of implementation, an exponential function, a sine,
something quite a long time to calculate the cosine field.
Or, we immediately we enter the small computer calculator at hand
but it turns out, our life leave us in an instant.
However, this is quite a Fourier series,
Pardon comes to account Taylor series; because you know cosine,
an infinite series, is an endless series of exponential functions.
This, we consider here,
You have to accounts in these grains each time and for each x value.
This function x instead of t be such that over time my speech.
Endless one for each moment, let not receive the infinite,
but you will account that a large number of exponential function.
But here, the exponential function once you'll accounts,
This beta; then you will get the length of this force,
so you will get slammed by multiplying multiplying itself.
This will be something extraordinary speed.
Instead account every time an infinite series,
You will receive an account number has been multiplied by one of their own.
This is something supremely fast.
For its turn,
this embodiment the fast Fourier series, called the fast Fourier transform.
English in "Fast Fourier Transform" it said and shortening called FFT.
Chemistry, physics lab who molecules, vibrations,
When they want to find something to vibration specificity, Or, a machine
part is the method used when you want to find the vibration of this.
Quickly give you the result.
Otherwise, this exponential functions grains,
Or, sines, cosines If ye would not account,
Despite anything to the computer's speed is too slow.
In the communication it is so useful that should be available continuously
To be me, you spoke, my voice taking the Fourier transform,
Please assume that I'm talking on the phone in my speech speed
that the Fourier transform of the sending of whether the counterparty account number,
the opposite side of the back of its Fourier coefficients
We have to be able to create function.
Here, if it can be used in this technology.
If not, it is used.
You are supposed to be many of your young age.
This computer, Skype and stuff out when it first appeared,
I would like vibrating vibrating,
because it could also convert these fast Fourier coefficients did not compute.
Now it has done fantastically fast.
For example, when you make calls with Skype numbers going to the opposite side.
Analog thing called my speech I,
take my picture, to send these images as faces.
This is a much more laborious and suffered on the road and also keep up appearances
not high resolution.
HDT, this is going to work their HIV HD High Definition television.
In Vehayut CD, the number of DVDs you do not bury there.
That number does not deteriorate; one is staying with a digit, the numbers remain the same.
Or, you sound machine that you are taking your TV, a fast
The Fourier transform, the Fourier transform of making sound or image.
This lies at the heart of this technology revolution.
As you can see, coming to a matrix multiplication.
Because the coefficients B's happening here on a number of factors,
B is developing a variety of forces.
It is no longer I do not want to turn the computer engineering course,
but we need to be aware of it.
We live in a great revolution.
This revolution that simple fast Fourier series and the Fourier series
It comes from a mind to be discovered.
Here it is stressed that with a little more, saying, digital,
audio, video, that's the same thing with digital digitalis e ...
digit finger means, in fact, comes from the thumb,
realization of audio and video and the IT revolution, the Fourier series
We owe say, a well as the FFT also rather fast
We owe the Fourier series will not mean much exaggerated.
Now for today, I pause here.
We saw four options.
A zero, zero Or, L range, the range has been given a function,
we direct it towards real, we can show the general Fourier series
or we can show with the series or its complex range of symmetric
L up to minus zero, or that the function symmetrically
as antisymmetric, we can show stretching.
You may ask: "What are its benefits?" He benefits
There are: General Fourier series earlier
Betting passed; A and B need to work with two different infinite series.
However, here or sinus, or you have the opportunity to work with kosinüsl.
This is an important thing.
E complex series as we have seen,
made with the complex exponential functions of the Fourier series,
It provides the opportunity to fast Fourier series.
That's not the sine and cosine directly.
For him, there use different place each of these four options.
Bye now.
After that we made, examples.
Here we will find several examples.
To make it easy just because a portion of accounts, a part of real life; a
how the vibrations of the machine parts are obtained,
What does the vibration of a molecule.
I, you, "Welcome." I say, what kind of a swing
and it is the Fourier transform, we will see them.
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