[BOŞ_SES] Hello. The previous section we saw our linear processors. The definition of this linear processor, then a linear concerning the definition of processor space, the target space, It reached the target in space, space, space transportation and definitions We also saw the concept of zero in the space space. We examine how these apply to the equation. If you remember the bet vectors in the plane, We started with a two-dimensional vector rate. We could do with this vector drawings, the collection of vectors We could do with geometric process, but progress is essential that When we refer to the number of geometrical sizes, When we saw that we can move cebirleştir geometry. And then draw the plane While it is possible come in three sizes, while theoretically possible practically difficult, When we passed next to impossible to proceed with the drawing in her size, but the number We can handle large size of three; even also in infinite dimensional spaces, their function could work in space. Now do the same plane, and planes in linear space We want to make our transition in linear processor. Here again we see that linear processor before, We are going to show the number of conversions. As they did so again showing the number of vectors in the plane and in the same way here also extends to the space that we do not speak, and so we can not draw large If we think we have made progress bi, where progress will be the same way. When we can show with great advances linear matrix processors You can also exceptionally easy to use the matrix on the computer. Therefore, they can limit the types of problems We have expanded fantastically. Now, let's start with these issues. Here, as we have said before, the only vectors When we were doing the numbers show significant progress. We can ask the same question, with similar ideas Is it possible to provide the display with the number of processors in a linear? How wherein when working with vectors that e1, e2 vectors already important, but also because of the presence of certain of them also We work with only the number you put them. Similarly, the target vector x and y in space We also show the vector with a new team. How can we show the numbers without using them? There is only one vector in the base, so we need his knowledge. Its also a single x1, x2 as the indicator was happening only. However, in the definition and conversion of space in a database, and the target space hence the two vector has a base, we are working with two vector sets. For an index future, each of the two index, We will work with number two indicators. When we fit them in a way that knee, A table is composed not stacking in one dimension we need two dimensions, See here because there is only one dimension is enough indicator, whereas when the two indicators We show in a two-dimensional table, called matrix in these statements. The term comes from Matrix where you thought, it also has an interesting historical background. When the first printing house was discovered, whether the letters from the lead even before they were doing the carving wood, After writing the articles they place them in a table, he was called matrix table in their knees, from there, to a two-dimensional table There are also words of a historical origins of this matrix. The Council also has something very simple. You brought a vector x to y with a conversion. Of course, the definition of space vector x wherein the base vector, The set of basis vectors e1, e2, so you get the n-dimensional y vector in the target space, Let them be m-dimensional space; In this target space, here's the bottom team in the F1, F2, FM say. n and m are not necessarily equal. They are more accurate, tidy if sigma summer collection If the icon with the XJ j j1 n becomes public until this tidy collection software. y also thinking of multiplication by a fine on it for the indication I'd post up to several meters as well say it again, we have shown collective tidy. Now let's write the ax. The effect here Axa's open software supplied x. But a linear transformation thus we know that as a result of the changing process. So instead of x with E for collection prior to and one hit if we return one, AE1, AE2, and they also çarpsak opposing the numbers x1, x2, xj, xn, we get it. Here we use a linear transformation. As you can see we write this statement in a tidy, i, j and there, a, a one by one basis vectors these must be acting; then take them in We collect multiply x with that left and right of the difference between the transaction as well. A total must be acting on this and also in it with x E has multiplied, here it is a individual must be acting on the e then we get hit with X collection. We're changing the order. This is an important consequence of the non-linear processor. DESCRIPTION moreover, the result of description. Here we observe viewed here X vector can be any vector x, but obvious these e e's. This means that a one by one to each unit vector carrying the target space. So let's say we take the aija typical one, The transport target them It can be expressed in terms of basis vectors of the space. Here j is it any good, because it is a base vector We expressed the base of the destination space. And j need this knowledge as you can see here There are both here for information, so that the coefficients overall a, i, j, for once, we can be good. So if we say we do all that is going on now, happening: ax effect on XJ times when e, i, j are there. Each e, i, j as well as showing how to gather it. Where e, i, j and we are writing. As you can see two collection occurs. The index is composed of two indicators. Now we change the order here, which in this collection because it is finite We gather we gather, so let's take out the collection on e Let's get on with the collection jar inside, The benefit of this is happening right: Here the collection i for here because it's all mixed together on. However, such as exchanging a, f, i we find on their own. We have allocated a total of happening here. However, for a combination of all here. There are separate. This is an important thing. Because we are set aside in terms of components. Again we compile this thing, it was ax y. AX we have calculated, by changing the order for the collection and we got out, we wrote that. y'ler ise yi, fi idi. Because this year in the target space, f de base vectors of the target space. E we now forgotten that there is a total head plug are denominated. Wherein up to several meters on the same total i, where the meters on the i. The components of the f y i say here, here it will be the sum f of the components. As you can see here anymore without the use of vectors x We have achieved year denominated. and coefficients in this definition one by one we see here floor space vector of the target in space We proved that came as the coefficient of conversion. Here this transformation, we are an equal number of years have expressed the transformation ax. This is the only issue here is how the x and y with just a number show we can, where we are treated at work in his results. Now we do not have transportation to our thereafter. Multiply the numbers completely, addition, multiplication, we can reach our goal with the collection process. This process is also very easy to do on the computer. O for calculations to be made on the computer matrix regard Want to technology, or sociological research to find the relationship, issues in basic science wants this matrix becomes the most useful tools. We demonstrate here. Now we will present how the knee in a jar? We know that the two index. Our ae1'i we have ae2'miz. Each of these m-dimensional space as well. Because of the number one for me. AE1 If we bring them in as a string columns, the work that we write to each coefficient in the first column, The number of first-E number; This comes as a second display. This place is the elements of the first column. The second indicator gives the number of columns here. This first column. The first line of the first column, second line, third line. The second column, always refer to the last two indices, second column E2 components, Does it will be one of them. Do we see that in one of these columns. The number of e gives the number of columns. So the definition was to have space, definition of the size of the space so that certain of these e It gives the number of columns in the matrix obtained for. Does the size of the target space was. This, we say sorry na, the number of the destination space, size gives the number of rows. Here also the juxtaposition of these two dimensions we call the matrix. Here is a term used in the printing It is coming. Now summarize, we give a definition for a base team space. We provide a base for the Target team space, It gives the relationship between them. Now this shows that any time we get j ai: ei j, e component on j fi. This also affects our thinking because each system Showing a vector event. J'yinc in vector output of the input vector i'yinc It shows the relationship between the coefficients aij. So this is not just a series of numbers, but also in our thoughts serving to build a structure. Here we tell you the number of rows and columns. If a greater number of aija, j'yinc means that the definition of space The incident appeared that the target space i'yinc there is a great correlation between the events. If this coefficient means quite independent from each other, these two events is small. That is to weak interactions, it's a table showing the strong interaction. With them it is easier because now we also make account Our progress with transformation typing long e f Zorken, We become just makes operations with numbers. Here right now if the number of e Is that the definition of space He m in size, because you have so many confident, is different from the size of the target space, we obtain a rectangular matrix. If we get a matrix it is equal to the square. Sometimes it happens so that both of these factors are both square matrices is nonzero only ones on the diagonal, while others are zero. These are extremely important, taking the diagonal matrix name. That may be a rectangular matrix, the matrix can be square. He gives the size of the number of the definition of column space. The number of lines gave the size of the target space. If two dimensions are equal, so the three-dimensional space to three-dimensional space Or, a matrix transformation would we do three three-pointers thousand When the matrix size would be equal to a thousand, remove a square matrix. We see that the number of columns equal in number because the line. Where the number of lines is equal to the number of columns. And as a very special structure, others only becomes zero on the diagonal is different from zero. This is an important structure. However, the matrix can be square. [BOŞ_SES] Now when given a vector, When given the floor and we have seen how this is a vector. We can find to balance the right and left components. And we can take the inner product. The same thing is possible in the matrix. Eija see now, i have convert j. This aija times f i output. Now here we do not know of a jar. j and f i e, we know, but we do not know of a jar. These matching, there are also prune the vector b, where b is the vector there, but impossible to find by matching them with internal cross to find that possible, if you take the dot product of this equation with f, in which a number. See here, a number that has for fk times. Therefore, we are turned vector algebra equation into the equation. From here, if the vector perpendicular bases If the perpendicular, here a just collection, only when k is equal to i will be a non-zero number. Then we will stay here aija fkfk times. So this right chord j j flow equation are you on the side FKF held by the inner product f k k is the square neck, When we divide, we can obtain in this way. Therefore, we can do both kinds. Completely matching the components of a vector, We will see them in the examples. Now we come to solving problems. Now we see, we see something very simple actually. If a transformation here, floor space in the individual definition We transform the vector, then we're writing for this transformation in terms of cost. To write in terms of these foundations, We need to find the equation for each of m and n is a number of times. One way to find this is to match the individual conversions, find another way in which the inner product. Now we take a break. After that we have achieved with solving problems We will try to reinforce this simple fact.