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.
Kare Matrisin Tersi / Inverse of a Square Matrix
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來自 Koç University 的課程
Doğrusal Cebir I: Uzaylar ve İşlemciler / Linear Algebra I: Spaces and Operators
25 個評分
Koç University
25 個評分
從本節課中
Matris İşlemleri / Matrix Operations
與講師見面
Attila AşkarProf. Dr.
Matematik Bölümü (Department of Mathematics)
[BOŞ_SES] Hello.
It has a rectangular previous session
We analyzed and we observed where this matrix
Although the left three inverse matrix
There was no reverse right and left also reverse the
In the matrix, the matrix could be the right vice versa.
Now there are some features in the square matrix, they have these privileges.
Theorem shows that: We prove this before, but in the future,
how to find the inverse of a square matrix that the second part of this course
We find that the criteria for reviewing the opposite.
If left to right inverse of a square matrix
We see that, conversely, is equal to that of the left and right matrix
We see and we see that only a single inverse matrix.
In contrast, we see that every square matrix, albeit not an opposite.
As I said here this yardstick, we will care,
When the criteria are the opposite, it requires the determinants,
I'm sure everyone has some knowledge about determinants.
But the issue we shall see in the second part of this course.
If there is one upside down
We show that a negative combined in a matrix.
That a merger of the right in a negative çarpsak,
çarpsak left in a minus combine, we get the identity matrix.
It had this privilege.
If a matrix or square matrix inverse of what we call the singular matrix.
If we say the opposite of regular matrix.
This is not very different from the numbers, but we know much wider in scope
Not a single case because the matrix is not a single matrix,
not the number of the number zero, not the opposite, but the opposite of a whole other issue.
Left there is also the opposite, right there opposite.
But the opposite is also possible.
We speak according to the product.
Now I start with an example.
A very simple matrix we choose the accounts to be easy.
two binary, one, two, minus one, get one.
Try to find the left to reverse this matrix.
If it is left to the matrix square matrix
two binary opposite to the left in order to be compatible
It supposed to be and that the unit is supposed to be the two binary matrix.
Now we do this multiplication, that will take the first column,
In a first column, we hit the first line.
We will take the inner product of two L1 L1 will be a minus.
Again we hit the first column to the second row L NSAIDs.
L2 L2 is a minus two.
This time we will pass to the second column,
Take the second column multiplied by two domestic first-line
L1 L1 plus two times here,
We will hit back with second column L1.
Here twice L2 plus
L2 two and it is supposed to be equal to the unit matrix.
There are four known now as you can see here.
A L1, L1 two, an L2, L2 two.
There are four of these provisions in the equation.
See, the first difference emerges here with a rectangular matrix.
In our example we have seen in a rectangular matrix of six unknown
against there were four equations, there are four against four unknown equation.
This is a positive thing and also once we solve these equations here
and also of the equation, we see that its in twos.
A L1, a combination of these two equations with two L1, L2 A,
L2, the second one with two equations, we gather here this equation
three times a L1, we find that there is no interest for one L1 L1 dilemma.
L1 L1 of one to two reveal itself immediately.
[BOŞ_SES] Here,
two in the second equation L2
for example, an L1, L2, we see that a.
If L2 L2 is a place where instead of two,
L2 would be a three, it gives you three in one over.
L2, we found that two of the three because of a reverse split
matrix matrix only once as we found out, and there he would be.
Now before you make a right, let's call right inverse matrix.
In order to do right from the inverse matrix,
two of R also needs to be a binary matrix.
This unit will get hit right matrix.
So we're gonna hit a two plus R is the first column,
We find here again the first column of the second line
We find a hit by a plus minus R R two.
Similarly R is a good first line gets hit with that,
R E is a two plus two times we found two one here,
again, this is the second column in the second row to hit the minus
R & R, we find a two-plus-two-two, this unit comes equal to the matrix.
Now, again, we have four against four unknown equation.
We also still easily solve these equations.
For example, a cooperative R & R two equations
cooperation here will come together with a third equation.
R & R is a binary equation with the fourth coming together of two binary equation.
We see here immediately, an R1, R2 is an equal.
Here we find one we put one of R1, R2.
R2 immediately turns, where two R2, R1, two vice versa,
thus situated, we find this number here again.
Since two of R1, R2, minus two times two,
here's another one three times minus two R2, it is happening.
Here are two of R2 to R1 two immediately.
When we write R as we have achieved here.
We see now where R is the same as the LAN.
See here a divided by three,
a minus two divided by three divided by three, see here a divided by three divided by three,
A divided three, minus two divided by three, one divided by three.
So our first observation of L and R are the same.
Now we look to make these two the same,
L to bring an end to A'yl right to multiply or multiply R and I are put,
We observed that both units given matrix.
Let's make one.
Then take the first column of the first sayır L we stood.
As you can see three plus one divided by two divided by three, he gave one.
Let's take the latter.
Multiply the first column to the second line, attachment,
plus one divided by a trailing three minus three, he gave zero.
Similarly, others also always gives the same result.
Thus if the mold frame from the left
If both opposite
There are equal and one.
Very different from the rectangular matrix.
Will recognize our example, we chose there a
three rows of rectangular two-column rectangular matrix
Matrix had left three contrary, had no right and vice versa.
But this privilege of square matrix B sample
but we saw a significant privilege over.
When it's because we do not want to have to answer the questions here
We need a determinant for the concept.
As an example let's do more.
This matrix of one half, two let a split of a matrix.
Otherwise, let's call it the left.
Yet a square matrix.
We see that when we multiply these two equations here.
A L1, L1 two,
L1 L2 combine two equations inconsistent with each other because both L1 L1 plus one,
L1 L1 will be two plus two times zero.
Because here the elements of the matrix elements of the right unit,
unit in the second element of the first row
We need to do this matrix is equivalent to zero.
When we found this right in half, it turns out the same as the second equation.
L1 L1 in a one plus two equals one,
L1 L1 plus two equals zero in the second.
Or, we remove them from each other, so I think with that I hit,
Suppose we remove from the equation.
On the left side will remain zero.
On the right side is equal to zero, ie two would remain a discrepancy as two.
Similarly, where there is a discrepancy.
So we can not find the left and vice versa.
Left has no opposite.
Here it is there right opposite might ask the same question.
Again, we take the same matrix.
Again, the equations are also inconsistent,
I do not want to go in because this is a very ordinary calculations much detail,
A'yl R. to hit the elements that we find here,
R1 will be a two plus two R a.
R1 is two plus two R will be zero.
So one of the same equation will be equal to one, one will be equal to zero.
Or, imagine that you remove them from each other.
Left remains zero.
It remains a right.
That is equal to zero as inconsistent.
Similarly, the same case in other equations.
Two R1, R2, the two should be equal to zero,
Also two of R1 R2 should be equal to two.
Because wherein the matrix is to be an equal right matrix
a whole elements, the elements must be equal in the same position.
Here again, you exit the equation from one another, it is equal to zero would be a minus.
We saw that again inconsistent.
R result means that no right opposite,
What's left to the opposite when we combine them both, what's right and vice versa.
So do not reverse the left, not right on the opposite.
Whereas in the previous example had left and vice versa,
There were also right opposite, and both were single were equal to each other.
This wealth out
We need a yardstick to take off when he found inverse of the matrix.
As I said, this concept requires determinants.
I'm also here have done a little advertising.
If you continue the second part will deal with there only square matrix.
We will also find other privileges of the square matrix.
All using them, we're going to the concept of a determinant.
I'm sure everyone heard again the determinant of the name,
maybe some of you even know how to do.
But why do we have to wait to see if the second part of the case.
We made a small example here.
But in general, we have seen threads,
the privileges of the rectangular matrix of square matrix.
Various assignments are given to you.
I encourage you to do so.
Homework regarding procedures, left and right stand
According to the search for such assignments are reversed.
So we finish one section.
What we see as we look at a bird called.
Here we see the basic concepts.
There are two main issues.
Linear spaces and linear processors.
Linear spaces within a space which examines events.
Linear processor also examines the relationship between space.
We've done a preparatory department before logging.
Making this tour d'horizon as the general topics
We tried to see the location of the linear algebra mathematics in general.
After that, what they have learned from the vector of the plane
We have the concept of using linear uzauy.
What we have learned a lot from the equation in two unknowns.
We have created an infrastructure in the linear processor with them.
After linear spaces and linear spaces here
After seeing the concept of algebraic spaces, or finite-dimensional spaces
We saw that things were more or less the same in infinite dimensional spaces.
As well as proceed to linear processor,
After reviewing relations between the two spaces before and that
before then define what linear processor,
the matrix of linear processor
We have studied on the basic operations can be shown in the matrix.
After seeing a summary of one of these sections in the second part of this lesson
We are looking at the determinants concept.
We're looking for examining the concept of inverse matrix determinant.
We see eigenvalues,
diagonal matrix which makes
vector and matrix vector o
When you hit a vector transforms its direction only.
With this regard, there is diagonalization.Lines event.
How do we make a diagonal matrix?
A matrix functions as described in this but nevertheless square matrix,
who knows something.
And with some simple differential matrix function
We will see how to solve the equation.
I hope you develop your knowledge reservoir
and more issues
You saw the look in abstract terms, but much more general terms.
Because the plane, which is the size of the vector operations
and even able to generalize the infinite dimensions.
Yet in two, so two equations in the unknown
Our observation can be applied to many matrix.
