Doğrusal cebir ikili dizinin ikincisi olan bu ders birinci derste verilen temel bilgilerin üzerine eklemeler yapılarak tamamen matris işlemleri ve uygulamalarını kapsamaktadır. Cebirsel denklem sistemleri, sonuçların tekilliği ve var olup olmadığı, determinantlar ve onların doğal olarak nasıl oluştuğu, öz değer problemleri ve onların matris fonksiyonlarına uygulanışı vb. konulara derste değinilmektedir. Ders gerçek yaşamdan gelen uygulamaları da tanıtmaya önem veren “içerikli yaklaşımla” tasarlanmıştır. Bölümler: Bölüm 1: Doğrusal Cebir I'in Özeti Bölüm 2: Kare Matrislerde Determinant Bölüm 3: Kare Matrislerin Tersi Bölüm 4: Kare Matrislerde Özdeğer Sorunu Bölüm 5: Matrislerin Köşegenleştirilmesi Bölüm 6: Matris Fonksiyonları Bölüm 7: Matrislerle Diferansiyel Denklem Takımları ----------- This second of the sequence of two courses builds on the fundamentals of the first course, is entirely on matrix algebra and applications. Specifically, the studies include systems of algebraic equations including the existence and uniqueness of solutions, determinants and how they arise naturally, eigenvalue problems with their applications to diagonalization and matrix functions. The course is designed in the same spirit as the first one 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: Summary of Linear Algebra I Chapter 2: Determinant Chapter 3: Inverse of Square Matrices Chapter 4: Eigenvalue Problem in Square Matrices Chapter 5: Diagonalization of Matrices Chapter 6: Matrix Functions Chapter 7: Matrices and Systems of Differential Equations ----------- 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.
Cramer Kuralı / Cramer's Rule
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En provenance du cours de Koç University
Doğrusal Cebir II: Kare Matrisler, Hesaplama Yöntemleri ve Uygulamalar / Linear Algebra II: Square Matrices, Calculation Methods and Applications
14 notes
À partir de la leçon
Kare Matrislerin Tersi / Inverse of Square Matrices
Rencontrer les enseignants
Attila AşkarProf. Dr.
Matematik Bölümü (Department of Mathematics)
[BOŞ_SES] Hello.
Cramer's rule number in the many small, little
number of unknowns in the solution of the equation is a supremely convenient system.
In general, it is true.
But the most commonly used where a small, small numbered equations.
Let's begin to implement it in two equations with two unknowns.
We see it, we saw many times.
The first equation we pass through a second hit minus merger
x1 is not,
minus a second equation 1 2'yl first
If we multiply the equation of a second 1-up x2
and here we see that we are not in the following description.
These are what we call the first determinant,
this latter a matrix consisting of coefficients of a matrix
Using the right hand side of the first column
obtained by the determinant of the matrix,
on the contrary, the right side of the matrix
martis used as the second column,
We show also deta2'yl determinant and also the ratio thereof,
if not zero determinant, we find x1 and x2.
We have seen the details on page 51.
Now we will apply them to the general matrix.
Generalization of the essence, expansions,
It is a new application of the Laplace expansion formula.
We have seen that the solution of equations Ax = b-1, to the left
If we multiply x times b is equal to -1,
ayrıca da A-1'in de cofA'nın devriğinin determinanta
We see that the division of a number of determinants, of course.
We divide this number.
Kofaktörl because the b-1, whereby such a product,
a number of division of the COF, we also when we hit b'yl matrix,
Here transpose b'yl multiplication of Coface.
If you write it in terms of components x
j'yinc component
See this COF
The first line of b'yl to çarpılmış.
We call we detaj.
So it turns out that the determinant of Ajman j'yinc line.
Such a conclusion we find generalized to the brew.
Underlying this underlying this solution,
Coface determinants of the fact that in the right-hand side
b'yl nothing more than to çarpılmış.
Hence, Cramer's rule, the two bilinmelenyenl
The two equations we see more clearly the rules generally apply as well.
We're doing it this way, given us a matrix.
We put it to bring the b j'yinc column.
Yet the divide, multiply N. consists of a matrix.
A number determinant of this matrix.
This number detaj say, he,
we find we get the formula just now.
Indeed, this sorry you throw a line in this column,
You also throw a j'yinc row, remains COF already.
That means, that we also detaj, j'yinc of this matrix
Place column b'yl to you nothing but the next expansion.
In this example, we can easily do.
Bi binary system.
Whether the right side, we find DeTai easily.
Six minus four to minus two,
We write deta1 right side instead of the first column.
We are writing a two instead of three.
We calculate this determinant.
Four minus four equals zero.
The right side of the deta2 we are writing instead of the second column.
A three remain the same.
The coming two two instead of four.
Here we calculate the time also, we find a negative.
When we divide deta1 of DETA, a zero.
When we divide deta2'y DeTai to minus one minus split in two,
giving it two in a split.
If we really look for a solution that he first equation x1
plus x2 x2 equals one.
x1 x2 represents a plus.
When we put zeros instead of x1, x2 also split a time instead of two, which we,
We find that it is really a.
Four x2 x1 plus two equals three second equation.
When we put zeros instead of x1, these three falls.
The two also split a time that we put in place x2,
Four divided by two as we observe it two equations provided.
If we want to solve the three equations with three unknowns,
We calculate the determinant again.
Right side instead of the first column
We account when we put the determinant of the matrix.
The second column in place, the right side, instead of the third column
The right side of the numbers we find the determinant obtained by sheep.
A1 A2 A3 to take these numbers
When we divide determinants minus two,
x1 divided by 12 minus two minus six,
X2 minus six minus two divided by three,
x3 minus four split, we find that two minus two.
Let's look at supply.
The first equation is x2 x1 plus two equals zero.
plus x2 x2 x1 equals zero.
six x2 x1 minus three minus six plus six equals zero.
The first equation is provided.
The second equation x1 not reset to the beginning.
Three plus four x2 x3 where our equation.
X2 right side of a three, so the first term of nine.
x2 minus two, minus eight second term.
Nine minus eight equals one.
The second equation also provided.
Similarly, we see that the third equation is provided.
Cramer's Rule, as I said,
A cofactor
that comes to the determination of b'yl multiplication,
they would determine that these determinants.
Now we finish this chapter.
Here I would advise you to do the following question.
They have done this before we
matrices.
We examine Martis.
We calculate these determinants as homework.
Now here is the presence of these reverse Laplace in expansion
sub-matrix, ie with the determinants and cofactors
Finding your way and in the second, with Gauss-Jordan elimination,
So write this matrix unit extending putting Matira
We expect to find with Gaussian elimination.
We also this summary here.
Here's what we found earlier formulas.
They're strong compile times by the transpose of the cofactor det
We see that time come to matrix units, and are taking account of here.
Matrix given.
This j'yinc column line to the opposite i'yinc
We say that the determinant of the sub-matrix Mij.
Mij said, no column j'yinc, i'yinc no line.
A matrix for each j, we find that it opposed the determinant.
We achieve this by placing an M matrix determinants.
This matrix M is in the plus or minus signs you change the cofactor.
Divided by taking the transpose of the cofactor determinants we find the inverse matrix.
The most important use of the inverse matrix where the x is equal to b of the equation
x is equal to multiplying the inverse matrix we see that b.
Likewise, the Gauss-Jordan elimination can be found and
Cramer we are treated in the rules.
Now that we are finished with this part here as well.
Still frame defined in the matrix in non-square matrix
We will look at the issue of eigenvalues and eigenvectors is undefined.
And at first it
where we will see what it means.
Eigenvalues and in the subsequent section
We will see their eigenvectors implementation of the concept.
Diagonalization of square matrix, one of them,
other square matrices of functions and differential square matrix
solution using the eigenvalues in the equation.
So here's a package.
The leader of the square matrices and eigenvalues and eigenvectors subject of this package.
After that we started this topic.
After you talk about this issue is that the three other basic applications.
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