Hello. Now I want to remind once the course structure. Because if so how well the subject instead of building our minds He sat down to easier. We are doing this course in two parts. First, basic concepts, and the second matrices, particularly applications related to the square matrix. This basic concept in linear first portion Our main topic spaces and linear processors. But we begin with an infrastructure like this: linear algebra with a tour d'horizon We have identified the place in general university education in mathematics. We talk a bit about the nature of mathematics. That may sound simple at first but then as the subject We started with two parts including all the main sub-topics. Vector in one plane, You know vectors in the plane able to draw here. Therefore possible to develop a önsezg. We determine the two numbers, we determine two aspects. As we will see in three dimensions and then some, but here, the large size of three and even be able to go in infinite dimensions. Functions going on in infinite dimension space. They can also be analyzed under linear spaces. We examined two variable equations and they also There is so much to teach as we approach the fundamental issues of a more abstract, Knowing behind them, yet abstract issues It is useful with regard to internalize and comprehend. Now that more direct linear algebra, we begin with a more abstract construction. The first issue of our linear spaces and linear applications of it as just a We will talk about the function of the space and Fourier series. In this section, the first section, section 4, algebraic spaces, we are concerned with the finite dimensional space. We also see that the basic concept of infinite dimensional spaces We will see can be applied. Fourier series are also experiencing a serious revolution today; Contact, digital communications, digital display issues. The essence of this method is the most commonly used Fourier series. After doing this, we define relations between the two linear space. This also happens with linear processor. In the first part of the space to a single linear linear space We are looking for dynamic. Here the relationship between the linear spaces. It will make it easier for examining applications We learn the way to show us the numbers, which also leads to the matrix. The matrix will finish with the dilemma of seeing some of these basic operations. Our sub-headings as follows: Now, part 4 linear spaces. We need to identify what was once the linear space. An important concept here and now that the linear independence going application of basis vectors. How to display the two-dimensional plane in the vectors i and j We used the unit vector; it is becoming more this generalization examine the linear independence and basis vectors. Turn on vector inner product with the how the plane here We learned calculation, of course, the three-dimensional space from large now we know we are losing the sense of geometric terms. But still it continues to be used as an important process. When we say all these things is the end Hilbert Spaces coming to a more complete description of the linear space space We'll see all of them at a certain pace. Now we move to the definition of linear space. Once we start with a cluster. If you remember, we started in the vectors in the plane with a set of points. Then a set of vectors, combining them to start occurred. We have also described transactions on this cluster. Actually, it was two of our basic operations. Collecting a number of two vectors and the product of a vector. We have the concept of linear space by defining this process. This is to establish a relationship between a set of objects We define the function. Here's more from its numerical functions genellenerek dependent and independent variables vectors to function in the generalized sense to be able to distinguish a bit or processor is also used in Turkish, we use the terms operator. You will recall, we can approach the three types of the vectors. A: 2 in size by drawing geometric path in the plane. This geometric definition of the two numbers We find its equivalent by determining the algebraic way. Now we define two basic operations with more generalized. Total operations of multiplication and two vector with a number. These are the basic proposition, the axiom or postulate, we use the term. We call those events to collect. The process by which these four propositions, It describes a collection. The first X Y X Y collection by collecting a proposition that recognizes that give the same result. The three elements, the three elements of the cluster, for us they will be generally vectors, before picking up the Y and X, Y of Z before adding it after picking it Xu Z. We accept that the same result of the addition. Both X for an impartial, neutral item and it is an element which changes the result when we added each XE. This neutral, neutral element. After you define the vector in one given this 0 The opposite vector, we define the reciprocal collection. We say the opposite of the first vector that have collected from 0 to her. There is a second set of propositions. This is a betayl you hit a vector, You multiply this number by a certain number, you multiply then again alfayl to. or you hit alfayl beta struck or X. is not struck by the number of similar procedures, did the collection. You collect alpha and beta. You hit the X or X vector to vector alfayl, Betayl hit the X vector, you collect. You collect two vectors again, after you hit the number one alpha. This X multiplying alfayl, and multiplying Y alfayl He says the results change when the two meet. Again there is a neutral element. This is the first number. A vector does not change the result when you multiply the number 1. These multiplication by a number, multiply by a scalar we say. This proposition is not a random structure. These are selected to reflect the events in our daily experience. Already when we consider vectors, we see that all these propositions provide. Thus these vectors can draw parallelogram proposition As we have defined as the direction of extension of its rules abstract downloaded more operations and we are very basic. This also linear algebra, what a basic training When we download the series as a simple proposition, A fine example of how powerful nature of what we have achieved. Judging by the nature of these propositions must collect a required number with the shock, Significant differences between the left and right sides as a change in one part. Here, the left and changes the order of operations performed on the right. So before you hit the betayl after the bump instead of a alfayl, alfayl beta to hit after hit vector. Similarly, in which there is such a proposition, It has a neutral propositions. Zero collector. A number of count with the product. This statement tells us that. The sequence of operations remains unchanged. In this way the proposition T1, T2 or T1 proposition in, C2 or C3 and proposition in a neutral element, so that it does not change the result when you add to any vector. We can define it in the collection by using the inverse of a vector. Here again, an inert, We define impartial process. Left and right when we hit the combine remains the same. As I said, from these two species. Some of T1 X. gather or collect x of y to y or y X. collect, Or, in other to make it to add a z and the multiplication This process shows that change results. the existence of a neutral element. We see that they also need. This process enables a process which these propositions collection, call number, providing a bump in the T and O provide. Now they are in these vectors. Already when we collect an x vector with a vector y, We're going to start where the last point. We are adding or adding a y x y x. This parallelogram go by a rule or go to the other side. We call this gives the same result. We are adding or y on x, x plus y is. We brought it adds y to z or z are adding on, We also added on the z x, then we arrive at the same point again. We take a vector or a product, we bumped it to a betayl. We stood in this emerging vector alfayl stands before them or to xi alfayl. Then we stood x betayl to. Both the alpha-beta x x and when we collect, here derived alpha plus beta x x, We see that alpha plus beta x is x. Similarly, a vector x, we hit a alfayl, we found it. The y alfayl we hit again, we find this vector. We gather these two. We collect or X. y before we hit out of alfayl to. Refer to hit alfayl after picking seen here before, gather here before hitting the alfayl. So we see that changing the order of results. Let's do an example. Just her Toplama and önerdiğimiz, Are we offer a collection process? get as x and y. They tolerated, as a rule, when I gather I want: Let me first component of the first vector, Let me be the second component of the second vector sum. Is this acceptable in a collection I wonder, is not? If the premise of the collection provides a collection T1 up to T4 it. If the collection is not provided. For example, T1, without blocking. He said that the T1, X. y, I gather that time, I collect a change in sequence. Let's see if the same thing will be? X. y when we collect our rule, we will get the first element of the first vector. We collect second elements. Where x is a stay in first. When we get in the other, our first year to now was no longer vector. We take the first element y. Second, we collect components for the second item. As you can see, although the second component first at the same lo components are not the same exit. Therefore, x plus y plus x and y are not the same. This means that we can not provide T1. We can deal with the others. I do not want to go here separately. This second We see that we provide T2 in an interesting way. You can look at it. But we see that T3 is also provided. If we take a zero-zero-zero elements, we get x one as the first item. The second element is picking up as we were getting two. As you can see that we provide. If he has collected by an X. y, y that we want to find a zero. What should be the x when given y would be the sum of the zero vector? So we want to find the inverse of the vector. If this is done, we get x plus y from x one. We get the first element. We take the difference for the second item. Now here we see that it is equal to zero. Taking x2 y2 minus, but are able to provide no information y1 and x1, forcing it to zero. Therefore, we can find an upside to all x vectors. We can reverse only the first element vectors to zero. It also shows a reverse T4 was not secured and we identify. Now if here that took the sum total of each item's If we give a definition, so alternatively, We saw just now that such a collection definition. The sum of the first component as the first item If we take the total of the second component as a second component, Now we do not go on it here, one that can not already T1 was that the change of x and y providing the order. We see here that provides. Because before y, if we will be again x1 plus y1. We could not find the reverse, here we see that Y1 If you receive minus x1, x2 y2 minus If you receive We see each other in a reverse can be found in the x. Therefore, this provides all propositions. This we know already parallelogram rule of vector in the plane. Again, as with alpha shock follows a simple process Let us define the alfayl to çarpılmış only the first component, get a zero for the latter. You may not like it, and in fact can be understood that this is a good process, but I'm also free to define it. What can cut my freedom? Providing these propositions can not provide. Now if you look at it, we need to provide as C1 to C4. For example, when we received an alpha, see what happens rules, You will then get a hit the first component of the merger, The second also says that I propose here will get zero, I proposed definition. This is not equal to the x itself. Thus defined, it would not be able to identify a number of neutral. Therefore as a product, it can not be multiplied by a number. But both components also found in alfayl to hit the vector, a number as we know we are identified with the current multiplication. For now, stay here, I want to stop. Because then we will see many more examples, but before What the proposition that an order for you to exercise yourselves, Thoroughly internalize recognition can be useful for a time.