Çözümlü Problemler / Solved Problems

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Doğrusal Cebir I: Uzaylar ve İşlemciler / Linear Algebra I: Spaces and Operators

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Koç University

Doğrusal Cebir I: Uzaylar ve İşlemciler / Linear Algebra I: Spaces and Operators

26 ratings

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.

From the lesson

Matris İşlemleri / Matrix Operations

Temel Matris İşlemleri / Basic Matrix Operations20:55

Çözümlü Problemler / Solved Problems6:41

Ters Matris: Genel İnceleme / Inverse Matrix: General Study17:55

Kare Matrisin Tersi / Inverse of a Square Matrix17:08

Meet the Instructors

Attila Aşkar
Prof. Dr.
Matematik Bölümü (Department of Mathematics)

[BOŞ_SES] Hello.

Our previous session, addition, multiplication by a number,

We saw operations such as taking the inverse addition of the two matrices to be multiplied.

We saw the compatibility to make this process also.

Now that the A and B matrices given us.

We want to find these matrices five times.

Then multiplied it five times to say all five of these items.

Here we stood all this is achieved.

Balls say, are we writing,

We are writing to B, collect items from opposing.

Here we see that B plus B plus.

Matrix multiplication are writing again,

We're taking the first column, we turn,

We stood in the first line, write the number that we are here.

For example, one of them hits reset on the first line,

i would stand out two and two, we stood three to zero, I come to that four of the two.

We rotate the stands to take the same line,

See minus two sounds.

Similarly to bring it horizontal took second column

We take the inner product of the first line.

See, the product of a merger of one, two plus three,

three multiplied six, three to be more the product of two six;

This gives us a number.

In this way, we obtain the matrix.

When we get to B, B to change the order A'yl

We stood; As you can see A'yl B's product is not equivalent to a multiplication by B.

This is an important feature that should be known once.

In some exceptional cases, they may be equal but their general rules

that it is not equal.

Even, in a kind of matrix identifies the class interesting.

But collecting the sum of the number of A'yl to B

For the same, because we're getting just collecting.

Collecting the change order,

The product generally can not be changed as well.

Let matrices A and B again.

B. A'yl to hit you get here,

it equiv who, let's face it ousted the account, so let's transpose account.

We will change the columns to rows.

As you can see, this matrix is being achieved here.

Let's do the opposite, so as to take ousted before multiply.

B when we hit with the inverse transpose

We see that B is equal to the inverse of the product A'yl.

EU in order to be aware of where things A'yl B; wherein

BA is going well now is overturned by crashing style.

We are treated with numbers in it.

Again, A and B matrices are provided.

This time frame matrix, not the previous ones were a square matrix.

These see the two lines

Occurs, B consists of three lines.

A consists of three columns, B consists of four columns.

A plus B is undefined, because of which every element

We will collect the B expressed in the new total.

So we can not find a B from each of the opposing Or, vice versa.

Thus, A plus B is undefined.

B is the number of rows to make the number of lines of the collection,

The number of columns to be equal to B with the number of columns.

Let's look at multiplication.

We are writing to before when we hit A'yl A to B, and then B to.

And receive the second matrix column

We take the inner product of the first line.

There are compatibility here to buy the first column,

buy, should be equal in number to the number of the second line.

We do this multiplication.

The results of the first row number,

the second the number of columns is determined yla.

But when we try to shock B'yl Au,

We turn here to take the second column in the matrix, which is incompatible intermediates that.

Therefore, this product is not equal to just BE, not even the definition.

Square matrices A and B if B matrix multiplication are available, but are not equal.

As we have seen here is multiplied by B, A and B also have multiplied.

Because these square matrix columns and rows in both identical.

But they do not come here just equal, but both defined.

But one rectangular matrix to define the öbürkü

It can be defined.

[BOŞ_SES] Now we need to take some special matrix hereinafter.

We have now treated as related to the matrix.

Now take a break before the special matrix

You need to grasp thoroughly and compatibility are passed through these examples,

as well as how to complete the process.

In particular shock,

a process that is perhaps a bit more attention from öbürkü.

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