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.
Matrisin Mertebesi / Rank of a Matrix
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From the course by Koç University
Doğrusal Cebir I: Uzaylar ve İşlemciler / Linear Algebra I: Spaces and Operators
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Koç University
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From the lesson
Doğrusal İşlemcilerden Matrislere Geçiş / From Linear Operators to Matrices
Meet the Instructors
Attila AşkarProf. Dr.
Matematik Bölümü (Department of Mathematics)
[BOŞ_SES] Hello.
Now a linear transformation,
A (x) = x and y given as a linear transformation
show with eija with an angled,
y are able to be obtained.
Due to the linear, A (x) 's,
i.e., it is shown here as y,
x, we see that there is a linear expansion of the product and a eija,
ie, a vector y in space transportation.
We also saw that these vectors AIJ matrix columns.
We see that a linear expansion in terms of the columns of a matrix.
But now there is the following question: Is the EJ are independent from each other?
And even though independent, is eija will be independent from each other?
This transformation
AEJ but this year it was being extracted from their linear expansion
The size of the space shuttle
So that the number of linearly independent of AIJ.
This means that the size of the matrix of the space shuttle,
The number of which is independent of the AEJ is clear.
We call on the size of the space shuttle in order.
Therefore, the order of the matrix transport
Does the size of the space, we say na.
Immediately following this theorem, following this definition,
The matrix of these Eiji
going column gives the number of the extent of matrix independent.
Now we have two questions here.
eija may be independent from each other.
eija may be independent from each other.
Here you need to extract them independently.
Our fundamental knowledge about these lies under the famous theorem.
The size of the space definition, the definition of this e-space
count up, so the number of x.
Not of the Independent, as the number of x.
That column of the matrix
Number of definition gives the size of the space.
But we know that the matrix is composed of column eija.
It also gives the number of which is independent of the size of the space shuttle.
Both are not equal, it turns out to challenge the size of the zero space.
In terms of matrices, we can interpret this in terms of the way line columns.
Without going too much on it,
As a theorem, we give proof.
The number of independent columns of a matrix of independent line
number are equal, that is possible.
Buddha. The number of independent column order
We identify as.
So, in order of the number of independent columns of the matrix,
It can be found in the number of independent lines.
There follows a description.
We have already seen an example of it right away.
Replacement of the two lines of a matrix,
multiplying the number of a row with a non-zero matrix,
It is trying to collect basic matrix operations of the other line.
They are already in the process of Gaussian elimination method.
Change the order of the two lines, with a line to hit a number.
Collect one line to the other, it constitutes the essence of Gaussian elimination.
Theorem says this: a basic matrix operations, matrix
It does not change the order.
So, initially the number of linearly independent of the line anyway,
At the end of this process independent process remains the same regardless of the number line.
Up on the theory the number of independent columns in the result or not.
Only this theorem we have proof.
Let's start with an example.
This matrix is given.
Now we want to find is independent of its rows and columns.
So we want to find the range.
We can do it with the Gaussian elimination process.
Matrix columns to s1, s2, s3, s4 say.
See, wherein one, two, three, there are four columns.
They struck the car eşitlesek zero, what is the solution of this linear bağılt?
If c1, c2, c3, c4 zero if possible, it's independent.
Or how many independent, then there is much order.
We can do something like this.
A'yl income equivalent to the product of the c.
See here c1, c2, c3, c4 to hit the,
Coming equivalent to the solution of this equation.
The right side of a simple equation of zero.
We can do this Gaussian elimination method.
Here we are writing here as matrix.
Reset the right side, putting you get the extended matrix.
Gaussian elimination method consists in three basic process you see already.
For example, the first line minus two plus hit,
If we add to the bottom line, we bring zero under the first element.
Others also changing accordingly.
The basis for Gauss elimination method,
bring on a diagonal keeping below the diagonal are zeros.
If we do this process, we arrive at a structure like this.
And here we change the location of the two lines as the last step,
We arrive at this structure completely.
As shown therein, the column of the matrix
The independent immediately apparent.
The first column independent of the second column.
He both independent of the third column.
But the fourth column of the first
the second column,
Pardon half of the second column of the third column
If you add a third column to take, we find this last fourth column.
So independent of the fourth column of the first column.
So, here we find three independent column.
We look at the number of independent lines, such lines this time
We need to get as vectors, going line by line vectors.
The first line, second line, third line.
As you can see, the fourth and fifth rows in a zero-zero-zero.
So, that does not consist of a non-empty vector.
As you can see, there are three columns in the independent,
There are also three independent line in.
Now here we go through the details of these processes.
Gaussian elimination accounts that you know them.
That these details are given here.
Where you come from, it is shown here.
So it's minus two plus hit, here it comes attaching zero.
First line second, attaching the third one comes under the zero line again.
The first line to hit the last line attaching the combined negative again comes to zero.
And in this way we make, we arrive at the structure we have achieved already.
What is important here to distinguish between dependent and independent lines of Gaussian
change the order of the basic processes in the screening process line.
The matrix to add to the other basic characteristics with a number one hit,
We know that change the properties of independence of rows and columns.
Now we want to stop here again.
Here we see the rank of a matrix, ie independent columns
We have seen the number is the same as the number of independent lines.
This gives the number of equations which are independent from each other in terms of equations.
Some equations are consistent but a useless zero equals zero,
We know that the information does not carry equations.
Let's pause again.
In the next module we learned that some issues
We will apply the equations.
In particular, we'll see how it plays a key role in the range of matrices.
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