[BOŞ_SES] Hello, preceding sections We examined the major subdivisions of linear algebra. These were linear space. Here, too, it starts from a set of linear space We have defined procedures in the cluster and in the end we came to the Hilbert space. Chapter four and five in the previous two sections ie We have dealt with the events of the dynamic within these spaces. Here we are concerned with the relationship between the two spaces. This is the second major subdivision linear algebra. Let's create the infrastructure so that initially a lesson We deal with that on the plane with two-dimensional vector, We are interested, we are interested in an equation in two unknowns. An equation of a conversion, a relationship between the known with unknown. A space so we think the unknown, known in another space It was also equations provide the relationship between the work we think this space. Linear processors continue to progress now here linear transformations used various names. We will examine relations between the two spaces from which the equation As I said a little while ago we went to the team because of an equation The relationship between the unknown known in another space in space. This relationship As we show the relationship between two spaces: a space composed of an x There is also space consisting of a year. Here's the tool that provides the relationship between x and y We call money processor operators, We call money conversion transformation, We call mapping sending money. These are the basic tools, including conceptual equivalent combination of small differences. In this way, the transition between spaces, As we can show the relationship of this space y y say x elements in an element of the contents elements the tool can show T into y. This is actually a generalized concept of a function, We're a relationship between the input and output in the function. The number of numerical functions x number of years. But here x the elements of a linear space consisting of many components, As it has elements of Rn function may also elements of space. Here, especially the one between Rn and Rn, which will examine the transformation of a space Rm. The relationship between functions then reinforce these concepts can also be seen in exactly the same context, Or, of course, the relationship between the functions integral differential equations We identify with the equations, their solutions in a separate thread. But as algebraic concept of a finite-dimensional space almost identical to the conversion into an infinite-dimensional space, therefore, a matter of infinite dimensions of course comes added, As we saw in Fourier series and Taylor series, almost the same concepts beyond that. Thus WILL learn much more extensive here It can be applied in areas. Let's start with a reminder function. We can see a function like this, we can see in different ways. When we give such a fine machine x x ' i y to return or change another description we can send a x y with a carrier and all that fi When we do get it is a function for x in the drawing. Indeed, here we take a series of x a y we take from the tables against them are also in this table Finally, though this process also shows us in discrete points. These x and y are very simple but very profound Dekart his comprehensive, intuition leads to a very important development, x take on an axis, opposing them also on another second axis y Taking a plane in space, we get a point in this case. After placing these points with a continuous line By combining them in a graph, we get the graph of the function. Yet opponents also comes with a conversion function. An incident here, here glum-looking human interest Take so we did, also used the theater to work in the transformation of a man smiling While this transformation as a symbol at the same time, give you an event The number of a vector, into an other things function. This concept functions. Now we can think of functions as follows: univariate functions of a numerical value to another numeric value but we can also work with combines functions of several variables. In fact, this series of lessons for multivariate functions series You can also look for korser. x and y are numbers may not vectors. the number of a vector in Rn formed, consisting of the number m Rm can provide this for the relationship between a vector space. This leads us to multivariate functions. Here I want to show to remind the function of several variables, linear algebra in order to determine the relationship with it. If we start with the simplest cases where arguments a single numerical variable can be a vector of unknowns. The output can be a vector dependent variables. To illustrate this more easily because the indicators for this index If you have a good idea if more than one variable. For his show at t x and y also showed the vector functions We're reaching for. Univariate vector function comes opposed to a curve in this space. If we leave it to the other side, that is dependent on a number of variables, Considering that the vector is an argument that the simple version of the argument consists of two numbers x and y. Y1 is also not much of a benefit should he carry this index Considering that we show our show with a surface in space. Of course they are, but they were able to further expand the simplest cases. We see the vector fields in a multivariate function. Here is composed of time with the XYZ position variable x is a variable often. These vectors u u u instead of in the dependent variable y wherein we use typically have physical elements, namely an electric field, speed, momentum heat flow, a magnetic field, it's physical size as a strength. This course can be seen in our previous lesson korsera. As relations with linear algebra; in linear algebra Does size still one of We think the relationship between the size of the vector in the vector x, where y and x but can get very generic, but f we take very special. f is composed of only linear function we thought We can only express any y, where x is the primary force of. They create the coefficients between the input and output Due to the need for the two index index If we get listed as a series of two-dimensional matrices of them. Therefore we are interested in our linear algebra sections in this chapter In fact a multivariate function relationships between elements in a space with n m elements in space you determine; but the simple function type, so now this one, Functions such as limits need, such as integration, without the need for concepts like derivatives, we can do only with algebraic operations. In our case, we move these issues. We will talk about the transformation from x to y. Let's start right here with the concept of a linear process. Linearity, remember that you were also in linear space. When we add these two vectors, in which we gather we collect, or even three, even to the one, in which we gather gather results did not change. Similarly, it did not matter when we hit with a number of the sequence. Linearity of the ordinary, made during the process If we interpret that as a result of independent here we meet with the same situation. See here a T processor, transformation, transmission, taking an x vector carries a y. Now, in doing so, just as well issue it goes like this: A vector We can convert or picking up someone else converts individual vectors, we collect there. Similarly, we convert a vector with a c number of hits. Or converts vector ago, we multiply the number c. Here significant difference between left and right, the order of operations. Transactions arguments here space is done in the left side, right side of the work, and conversion is provided; On the right side before conversion is provided, ie these vectors occurs in the transport space, We collect them here, similar to here. If a processor if the processor makes this feature we call it linear. If it does not provide non-linear or "nonlinear operator" call. Combining these two properties we can use. So multiply the number and then collecting event, We're here for more than two operations; We stood by the numbers, we are collecting. Similar as before, x and y vectors convert, hit them with numbers, We can also collect; We can generalize even more so. with the number one hit convert vector collection of individual transformation is equivalent to multiplying those numbers. In order to make it more compact, Show the sigma sign because we could collect, We collect the ck, you say them to each index k, We get left picking up multiple n. On the right is any, its typical one including the UK hit by the opposing number ck, We provide the collection with the sigma icon again. So we see the importance of linear processor in summary, but the results are not changed during the process of definition. Let's make a visual here. X and y are vectors in a space, we can collect them, x plus y occur. We convert it. Or convert x, going tx; We convert y, nd happening. So this t affect them separately in each. After gathering we do in this space. If these two vectors are equal, we convert after picking ago Before we collect or convert vectors are equal, We call this kind of processor-linear. The essence of the event does not change as a result of the transaction. Let's give an example of a non-linear as well. Again we collect X. y and we transform it. He came here. We are converting the x, y we convert to, these are the tx and ty, we collect them. As you can see here that the results depend on the process as well. t x plus y by applying, The collection of individual x and y t can not be applied to give the same results. This example of a non-linear processor. Of course there can be a lot easier in the sense of linear and might be more effective, as well as changing when results If the changing business means there will be more complicated. We start with linear processor because they are both more simple, but to model many phenomena in our linear We model the relationships. Let's try to give two examples to reinforce it. Then came a little more detail this u We will examine the characteristics of the space it is alien to t take it. You should be given a conversion. This transformation is a three dimensional vector; U1, U2, U3 is the vector with components, a space which carries two components. See here a component, the second component. Is this not a linear transformation manager now I look at him. This sequence as a vector space vector v Let us also, each of the two component, three component. v u'yl to multiply that out, we knew it straight We carry the sum of the components in space. Let's turn this total. When we convert now no longer occurs first item here u1 v1. U3 U2 V2 V3 that occurs, first second and third elements of the vector sum. Conversion tells us: Get the first element, the second element to collect, let's build the first item reached space. Because here we have the space for a two-dimensional space. The second element in this transformation, took the second component of the vector says collect the third component. Our second component here, our third component here, we collect them. Turn it into a separately u, v to turn it into, let's get them. When we first collect them from the same conversions and second u we take the elements; We take the second and third elements. V Similarly, first and second, and third elements of the second and you're getting. E sum of these two vectors in said first component wherein the sum of the first component. The second component of the second vector, wherein the second component sum, we see. In this collection of numbers, each one number, u2 U3 U1 V1 V2 V3, a number of them. There is no order of importance in the collection of numbers. The only difference here see U1 V1, u2 v2 We see a slightly different way, but the result is the same exit. We see that the second vector composed of the same components. So before changing after picking, to convert; After converting to collect it gives the same result as before. That means it makes the first rule of being a linear transformation. The second rule is to transform u hit by a car. C of a u'yl mean to multiply, c means to multiply all components. Now the conversion terms of these new components, It says the first and second balls received; c1 plus Cu2, that c is constant, Abilmesiyle taken out as a common term can be written in this way, wherein the second component likewise. But the ct (u) 2 starting as different; Before going into u, u components, we are going to two-dimensional space of a three-dimensional space, then bring them crashing C using, see here in completely different processes. But a vector, we know from linear space, all components of the means to multiply by a number of means to multiply by the same number. We reach the same conclusion as you see it again. So this T number with the product Also different from the result of the operation It shows that, that provide this feature, thus a linear processor for providing both the TE feature. One, let's start with a simple example again. The sample still being very little difference before, the U1, U2, Let's get a two-dimensional space as a three-dimensional space U3 again that just now, but Let him holding an additional second component is the same as the previous example. Now u and v vectors You are given three-dimensional space, we will convert to collect them. When we convert them, of course, the first component of its simple collection among them, the second, the third, we will take themselves. Simple multiplication by a number, multiplying the number of all the components of any vector of the means to multiply. Now here u plus v Let's turn. Conversions of u plus v is the sum of the first and second components wherein, because the rules give him, he says collecting the first and second components. Collect the second case the second and third components, He says adding an end. We apply this rule. Now we apply the same rules in this transformation u and v for u and v, that collect the first and second components, the second and third components We collect the adds, we are also there for similar transactions. This we see that adding two vectors are getting the same first elements left, second elements are the same tactics quite some time, but a positive one because here comes two. Even though these two vectors are not equal so close together as you can see, because here is where the two accumulated retained earnings. Thus, although the components of u and v is Even more improbable, the first force component forces, T so that a plus, u plus v is T'yl A transform T (u) plus T (v) to have different, so that the T is not a linear processor and enough on its own. Needless to say the testing process, Let's just habit, but also to see and our hands. CU knew, we transform it into T'yl, here likewise the sum of the first and second components, We are added to a sum of the second and third components. Similarly, before we turn u, In this definition the same as the first and as you can see here the sum of the second sum of the second and third elements plus one. C using it after when we hit on, we're looking just like the above, but also to be multiplied by C using, for always there will be a car, We see that these two results are different from each other. So this in itself is not linear, It is sufficient to demonstrate the transformation described herein. Let's look at something that we are also more familiar. y a size T from a one-dimensional space dimensional space conversion, we know that ordinary digital functions. Do you square that you get that x x x T channels it into squares. x1'l to collect and convert it means we x1 x2, where that element, that means you get the x1 plus x2 squared, we know that these be squared. When we convert single x1 and x2 gives the T x1 squared, T this time gives x2 squared. We see that again, we know the equation is not true that traditionally, a parabola of the equation of a curve, We already know a little bit here straight to where you come from, We understand that the forces appearing in a correct equation. We see here the first feature not available for that force. Similarly, if we dropped the second feature, on the CX When we apply the frame of the CX T will occur. The CX squared C squared, x squared. Before T, before we convert x T x squared occurs. As you can see we hit it in the car T CX conversion occurs in the c squared x squared, CX frame occurs here, so these two are not equal. Yet each of them, this transformation, where we know a numerical function, we prove something that we know to be linear, So what we know of this definition is consistent with the earlier, An example shows that it is compatible. Similarly, the derivative transactions with D Although we take the derivative of a function given a new function gives you. We know that this processor is because you çarpsa with a linear function, Do a b çarpsa with another function you take the derivative thereof, a'yl to çarpılmış of the first derivatives, We knew it would be b'yl to çarpılmış of the second derivative, derivative transactions as well as a means to transform processor we can look and see from there it was linear. Similarly, the integration process, whether defined get definite integrals, i.e. a, b across boundaries, whether indefinite integral by a certain x, Leaving aside for now the integral constant, I do not there is actually a mess, this is also a function of the heat, gives you a new function, such as taking x x squared divided by two giving, It is minus cosine sine giving a processor providing such a transformation. Recently those no difference. This function is effective in linear space but also because we see that it is with a function, the number çarpsak, We are calculating the integral hit by another b it is about the individual integration of these functions We know that the future of the çarpılmış with equal numbers a and b. So far we have encountered many issues as linear processor It emerges, similar to the non-linear, When such a conversion function is interpreted as allowing a curve We see that in providing the features of its linearity. Now it's time to take a break here, Let what we have learned and this new review, We go to more detailed information.