[BOŞ_SES] Hello. Our main topics in linear algebra and matrix A and two courses. Linear algebra tells the basic structure leading to a matrix. Linear spaces and linear processors. But we're also doing some simple calculations associated with matrices. In the second part, linear algebra We see the process related to two entirely matrix. Equations, determinants, eigenvalues of matrix inverse matrix, eigenvectors, We see fundamental issues such as diagonalization of the matrix. Of course the substance of these issues, not able to do calculations without understanding the basics. Furthermore, as a MATLAB software that is currently most effective, the most widely used software. This supremely effective software to make calculations about the matrix. Therefore, there is also interest in learning it. Therefore, what you need to define the basic structure to be solved, MATLAB also need to be resolved, but how it will complement our strength in our muscle so we can have an important tool to make this work. MATLAB word here, as we see 'Matrix laboratory 'made of shortened words. The main purpose of a software matrix. Already a computer can not constantly definition, working with discrete values. Whether for his continuous functions, these integrals, basic derivatives such as differential equations problems already solved rotated matrix. He respects the MATLAB software that can work in all areas. This is happening as many program entries, but this is also happening now is an interactive software output. That is an interactive software. You write the answer comes immediately. Maybe too big a problem though slightly open your eyes a few times You may have to close, but here is simple, without time to open and close your eyes, Coming answers to short problems. The main structure of you here, we see the theoretical structures linear algebra and linear algebra problems and a Two classes of analytically in the main text We will solve the problems we solve using the MATLAB software. A reminder: the Turkish MATLAB written in English for the original three, S, G, O, U, they, and letters can not be used as their capital letters. But this does not remove the difficulty. You'll see. The first step description of how it is done. Matrices, vectors, matrices, or vectors row column matrix structure can be and we want to start with matrix operations. Our first job is a how it is written of the line matrix. When the gap by writing primarily accepts this as a new number. Number five, minus three points, on a number. Gaps are important. This also shows that it is a comma overhead line. When you wrote that it appeared as follows: A line matrix As for the gap is. The first element of the row, the second element, the third element. Of course, it is just the opposite of this line matrix column matrix. The basis of a matrix column matrix. The first line, you spend it with a semicolon. See where it says when you put the line into space. When you put a semicolon in column writing. Five of writing as the first item in the column. After the number again and point gap separated by commas as the second element in the column writes and writes on the third item. These rows and columns. Same thing will happen in the matrix. See that line here as well As well as the existing line matrix column matrix composition. Gaps are taking elements of the line. One, two, minus three matrix of the first, second, is the third item. You reserve a semicolon second line, Disconnect the semicolon see you again as a third line. Writing is one of the issues that we need the most basic learning matrix. Now this point, another use of the matrix transpose the hill immediately comma, column matrix that is the transpose knock it, doing line matrix. Line matrix knock it, column matrix line so doing turning column line column. Here on the hill Or, comma sign indicates that this is the base line. When you remove the column that was it. Now that r hill If you check this line matrix commas so this line is the inverse matrix, the matrix makes columns. Numbers written the same but the column matrix. So we learned that one of the processes equal and semicolon Or, the base of this hill outside the comma sign. If you receive similar üssülü and s, s see where the column matrix. This üssülü, its transpose it knocks it writes the line matrix. Equal follows, taking you see this right, describes it as r He says right here that describes itself as s. Here he describes as a right summer. Or, as a third in the second matrix to define b. See again one, two, three, four, five, six, seven, eight, nine. This is the first line, second line, third line. Semicolon separates the rows. Gaps in the line separating the elements. See exactly where b is one, two, three, four, five, six, seven, eight, written nine. However, when we put the base account is the matrix transpose. Lines column, replacing the columns line. B can also be the same thing as TB base You are reminded that t he transposed he transposed where t're remembered. You are completely free to make definitions on the left. However, this software has its own unique icon. For example, where would you put in square brackets instead of parentheses normal, It does not accept, understand something. If you want to write matrix brackets. This is because it is something that can easily remain in mind. He üssülü B's account. We write this as TB. See here. Now this may come immediately to mind: We know that with a symmetric matrix transpose, equal to the matrix transpose. So if we remove the car from a tau, if the symmetric matrix C becomes zero matrix. Because a and t equal to each other. If not, it would not be zero. To control this both a and d are symmetrical d which define whether a well. b equals minus the TB d say. You see, see that car gave zero. This intermediate process we do not see anyway. Matrix them, giving us enjoying the software matrix. Looking to D is not zero. Just understand here A symmetric matrix with the difference that has since ousted zero. B matrix is not symmetric. The first they learned processes. A second type is multiplied by a number of transactions related to matrices, a dull, collecting two matrices, in particular two multiplying a matrix. We know that the matrix of a plus b is equal to b plus a matrix. But a times b matrices, it is not equal to B times. Here I want to draw more attention to something new. A title that gives the dispersant. This made the process of what we might want to remember that. Then we write such a title. A sample matrix operations. Sure it is not evil and even dotted the i in English. These 'examples' he writes, 'process' he writes. But it's not a problem being here also understand. Disperse the word that comes from the display. The display them, says you type them. When you give this as input, as it turned out, as the title says it already. You remember that time that's what you belong to. Matrix we have in the same way. First row from the second row, separated by semicolons, semicolon separated from the third row. See, wherein one, two, three, zero, minus one, four, minus two, one, zero. B is also where the first line, second line, third line. A number equivalent to multiplying by a matrix that is multiplied by the number of all elements. Now he is doing in all these processes. Making process is the right one, we do not see, giving C. Here also is collecting A'yl B., we do not see how it collects, It gives us as Dr. Similarly, A'yl B. the stands, B'yl AU stands, and gives us as E and F. They entered our author MATLAB is, of course, this is my entry to remind you that I gave this matter. Or you do not need to write it in MATLAB. Now we have here are. Multiplied to feed the feeder means wherein each element multiplying means. As you can see, each of the five to hit the state. Was hit by five to one, was hit by two of the feeder, the feeder was hit three, he wrote. And we also we do not see this process, we see the result. D is, A'yl B. is appreciated. The elements of the gathering of the B elements. Birla collecting zero, a writing. IR collect one effect, three writes. Trinity minus one gathering, two writes. Like this. A'yl B's product, as you can see the multiplication sign, It does not make our hands like a normal x It shows the impact of this star sign. B. You see giving as E. BA gives F and here's the next one as well as other general We see that in theory we saw in the order of matrix multiplication change the result. In their collection, A plus B plus B would be equal to the gathering because after these numbers. We're taking the first column B we invest, we knock, Then we collect hit by the opposing elements in the first row. And so we find E from this team. Indeed, one of them stood out reset. I stood two plus two is four. Reset three of them stood out, it was zero for four. Brought here he writes. These, of course, we do not see. Already there is a place in this force, We do not need our vision operations, giving the result. Yet we continue the process, we know that. Matrix with the matrix B hit, it overturned his, we get the transpose, A B'yl equal to the product of the coming revolution. We know that the main theory of this product in reverse order. We've given are, it was a B.. A'yl B. We stood here. Now is the inverse of B, we do not see. Then take the inverse, we do not see. We write Whether we like here, give it, give it said also. But need we did not want it because it is not required. This makes the product, as you can see these two equal. Equal to the inverse transpose of the product in the product of the coming revolution. We know that from the theory. Here we are in here have done a checksum. A third embodiment. Here, the difference The size of the matrix B is not the same with martin. See matrix has rows of three elements; one two Three. This is the second line. That two-line, three-column matrix. The B, see, there are four elements in the row. Four-column matrix but three We see here that line. We now know that, if we want to hit the A'yl B., We will receive the B column, we will transfer, will hit him with the number coming equivalent. B. In order to perform this operation The number of rows of columns has to be so. It has to be compliance with this and it's also the resulting matrix line up and B columns so that there must be a matrix. We stood A'yl B. is really a two-line, so the number of rows, four columns, we find a matrix B, the number of columns. But when we hit B'yl Au, We will see in the column, we will transfer, will hit the B line. So there are two elements to the column, there are four elements to the B line. This product is not compatible. The software also makes it to distinguish, so you come to this discordant information. He says it in English. But it also alerts you if this product. An important type of problem, solution of equations. AX get an equation r equals style. A matrix, a matrix three three-pointers in here. The same problem was solved in the text. Both in the first and second portions. X unknown, R on the right side. We given on the right side. See where the base is important because it needs to get the r column. In three rows and three columns of a matrix, the first row, dizeliy them as second line third line. So, here it brings consecutive matrix rows are side by side. Here also we want to solve, AXA is equal to r. Here dispersed in part, that display in the section we also exhibit in quotation marks and commas in the hills of this We say we want because what you write in the future as well because remember. It gives the output. We're following section. A backslash r. When he saw it, as we did in the Gaussian elimination, the coefficient matrix, When we combine excellent cleaning efficiency, we find the expanded matrix. The solution is doing, yet we do not see that the Gaussian elimination. He's doing inside, eventually bringing Xu says. We want to find a determinant here is because often the determinant something that was calculated. Det matrix brackets are typed into account the determinants saying. To do this we want to. Is different from zero determinant of a square matrix are the only solution. If zero, or would not be any solution or an infinite number of solutions. Indeed, as you can see here one turns out negative determinants in the first instance. The second and third samples determinant will be zero. One of them will be no solution, there will be endless in one solution. This MATLAB are given to you when you write them. They give it line by line. We are here because we do not fit in one place. Writes the matrix below equals says, writes matrix r is below says, Cleansers equals minus one, is account, we do not do it. So you can control by account, of course. And he solved here in bringing X writes the column. But also in our method of Gaussian elimination, such as last summer. So, we have learned two things here. When we wrote to say det brackets If you reckon the determinants of what makes her account. You are free to name the left side, you can put what you want. You could also say that a number of D but remind so we're also easy to get det. This dete your choice. det (A) which has its own language MATLAB but it should be defined in order to make this process of the course. When we say a split run, if you notice sign backslash, You have to write it, otherwise it will not accept. Coefficients to be run right here, here's the solution as X writes and writes it took away here. And we see that the only solution because we calculate the determinant. We must not do that. Or, just because we have something we prefer to control this issue I prefer something here. In the second example we will solve r equals AX again. We call this summer. We solved it by Gaussian elimination in the second instance he writes output. but before you write before you write it, Are you writing r, calculates you say that, you calculate it says, then you have a command run, run, run, lest çalıştıy sense to run the program and it gives you the output of grains. See, here are a few changes from the previous example, six five points in the previous example here, öbürkü are all the same, right side of the same; Five minus three, ten. Of course, changing the number zero determinant of zero to do this because it previously held an account. It does not always bring the determinant to zero to change a number. Just giving these outcomes, Last summer, the sight of our desire to control the determinant is zero, we know ourselves of this equation now, the only solution of this equation There is no, or has a zero solution or has infinite solutions. As you can see here, there is no solution. Inf, because here also it comes from infinity, The last line says is eternal because it goes like this. If you look at it you'll see that instead of text, zero, zero, going to zero as a result of Gaussian elimination. If the right-hand side becomes a non-zero number. The number of non-zero interest forever, of course, when this division by zero. Take it out when approaching a top forever. Not available here in the sense, that is no sense solution to this report. Seeing already detours, even if you have been calculating the determinant, You'll understand when you see it is not a solution. You may wonder then, why he did this solution. The sight of the general theory as determinant in calculating taught us, We understand that solution. In the third example again we solve an equation such as AX is equal to r. The coefficient matrix is the same as the previous one. But we changed a little bit right. We have changed the last number. Here are five minus three, while ten, leaving it exactly the same, minus twenty-one did, of course, Because this previously little studied, we see that twenty minus merger work. We say fix it again. We call on the one hand, calculate the determinant is not necessary. But that's just because we hold on to our issues under control writes to you, r he writes later recall. Determinant is account, we see that zero. We understand that when zero, or there is no solution or an infinite number of solutions. As you can see here, giving a solution, the latter is zero. This is one of the infinite solutions. This is a special solution. If you add any solution of this null space, you'll find a general solution. Do not make your way there, but I did not bother to think so. Why from now, you will see in the main text, the same sample. This last line of this Gaussian elimination goes off zero, zero, zero is happening. But also the elimination of the term on the right side, in the extended matrix, there you'll find a zero, then it is equal to zero is zero, no solution at all. But if you zero in this solution, One solution to this endless X and y, ie X can find the first and second variables. Give me something else, but not zero, you would find another solution. That gives you a clue to find the general solution, this is very important. You can find general solutions using it. If you look at the text, main text, both the first and second linear How to be understood that the algebra part of the overall solution. The presence of the inverse matrix associated with the matrix is an important issue. Now we have three examples here. We also know that there is no determinant of the inverse matrix is zero. Is not zero, there are opposite and unique. Now we describe here a B matrix. This B matrix is not random, but in fact you can define anything. This is the first example of the matrix of Gaussian elimination. We call detours, we want to calculate the determinant, we have to calculations, we have to use a definition. What we put into brackets calculates its determinant det. We put it to the previous one. He calculates the determinant of a given matrix. Here we put the B matrix, just to get a change, So what accounts to show that if we put him. Here's calculate the B determinant. As I said before, you can put the name you want to here left. But can such detb written for the reminder, not because you put parentheses within parentheses that det In that letter the authority of MATLAB, in his language. The reverse calculation of a matrix describes the inv'l. How is that, when you put it into the det B, if B calculate the determinant, When the inverse, calculate the inverse of the matrix in brackets. inverse inverse also comes from the English word means the opposite. To ensure the control of a gene event, we can make those products. B'yl B's when we hit the opposite, we need to get the identity matrix, If done correctly if accounts. This is a comforting thing, or nothing terms. In this article we have given our dispersed as well. This word comes from the display again. Parentheses, commas important this hill, base marks. Because I have the letter, he says to no English but we understand. And after giving the same article. We gave B. B. after giving, we also calculate the BI, BI computing. We also calculate the determinant, computing the determinant. It had a small range of changes here writing. Because we want before detb'y, brought here before detb'y author. He relaxes in us, because it is not zero, because we understand that there is the opposite. And inverse B'yl BI B, B You have to go in the opposite when it hits the identity matrix, if done right. It also stands, we do not see any of them again. And as you can see, the case reached the identity matrix. This matrix has the opposite. The determinant is already zero, not zero, He assured us that the opposite. Now we're getting a matrix as follows. This is the second example, the matrix in the third example. We call it calculates the determinant, will be the opposite and let's see We call calculate the inverse, call accounts and vice versa. And after this title also writes here. He does not say the portion of dispersant, braces do not write, just this hill comma, after writing between the base of the marks, B. writes. B is the determinant, as detb writes, comes to zero. Then we know that the main theory, if the determinant is zero, no matrix inverse. Too much already, MATLAB B's opposite so you do not give a statement in English. Then you already know that the opposite is also here. Let's do a little more for a large matrix. Of course, when three three-pointers, two binary when it is possible to do many things by hand. Able to make the stanza. We can do very well with Gaussian elimination Or, the opposite of the cofactor. Already selected a simple matrix, see you hungry, according to the fourth column, You got three three-pointers in a matrix calculation. Easy, is not a very difficult task. To give an example, here we are again giving matrix, There are four elements to see inside the line and four in a row. Here is a string successive rows. Matrix writing is writing in our language we understand, in our view we understand, The first line, second line, third line, the fourth line gives. This is the determinant account. This coincidence minus that of a prior course of a rise. Minus one turns out, that means that we know of here, this matrix will be the opposite. The reverse of the matrix inv'l, We know the word in the opposite sense from the inverse word. Nor could you say here, but to remind CI If you want from CT C in the opposite sense. If you want to Türkçeleştirme thoroughly but it also shows the relationship inv'l to, Coming out of here immediately calculate. And again, whether true product C using CI hit the unit in terms of providing if you need to get right up that if this inverse matrix. He gives as you hit D. And as you can see here the unit off matrix. We can do a little in hand. Get this example, the first column of the cycle, Multiply C through the first line. Two times four equals eight. Once minus five, minus five. That leaves three. We hit two of Birla minus, minus two, that leaves one. We crossed 11'l minus zero output. This second line after the çarpsak, see here four times, four. Reset will make five contributions. Two times minus two, zero. Minus two equals four. Here were four. Minus four also more zero. Again, this is happening given the zero. Already machine does not leave them need us to do. We do not provide in order to understand the dynamics of the sheer event. Now the most important in the matrix again One of the core values and essence of the issues subject vectors. In English to the core values and eigenvalue eigenvector, eigenvalue said. Extract meaning from German into a special meaning to the word. Eigen has been accepted impressions here. self eigen values of the matrix you put brackets Type and it gives you the self-vector. Select the row before giving eigenvectors, of course, will be more self-vector matrix, Q writes them as columns of a matrix. Q is a term we chose here. Q Why? Q. We agreed because the main text as well as the matrix of eigenvectors. In the lamp again, this is what we choose, because self-worth a little more a universal display, a lamp, two lambda lambda were shown as three. It is doing this: we're not here to matrix. This takes the matrix of self-worth and self-vectors, self-giving as the column vectors of a matrix Q, The core values from this matrix diagonalization matrix gives the numbers on the diagonal. If you see this text into exactly this problem was solved in the second part. Eigenvalues at two, four, six, he says. This order by choosing the little self. We have the right sort of small to large, two, four, was lower. Here, two, six, four. Now that you have come so that internal dynamics. Core values is giving way. The core essence of the first column vector in two opposing vectors Placing second column, six opposing eigenvector, The third column including extracts from four opposing vectors. Here follows inspections, we can provide. This matrix Q if it goes zero determinant, This means that dependent on three columns from each other. Three different core values is to be opposed to each We know from the general theory of self that is independent of the vector. So it's got to come out zero determinant. We've also calculated that here, calculate detq'y. detq'y has calculated, found minus two. We said, "Oh, okay, right" because we say these three discrete from each other would be self-independent vector for three core values are different, determinant of the matrix formed by them can not be zero. This determinant is not zero there is also the reverse of that Q. Here again, inv (Q), but then found the Q inverse, found the opposite. She finds. If you multiply the two units should come in as an inspection. We call it also made for monitoring, indeed the unit is off. Some of them in terms of our training, we did for ourselves thoroughly penetrates us to digest. Or does it wrong machine. Coefficients here, if you look into the text KOKI root of two divided by two. Stem two divided by two. Or, of course, root root here does two write two split two, this is equivalent to the number of writes. Of course the root of two is not a rational number, ie, infinite many digits after the decimal point, and we've got four of them chose to take. 12 one could also ask, as one could wish, You could also want two, yuvarlatarak we writers here. In all of this product because it is also the fact that some rounding He could be 0.9999. This resets 0.0001 could emerge. These would be considered minor rounding will bring. But this product is provided with the main line. A second sample, a sample which is still soluble in the text. Substances were given first line, second line, third line, He writes that brought consecutive rows. We say it again that eigen (A) The eigen values, find the eigen vectors. Core values it as Lambda, eigen values written on the diagonal. Important difference here is the previous example, a core value to be repeated. We know from the general theory that when repeated eigenvalues, two self worth repeating here twice. By contrast, you can not extract from two vectors can be. When we can not control what happens when you still here, because the matrix Q in this case we selected, what you want, If you want you can put your initials in the name. The first column writes eigenvectors from five opponents, He wrote one of the core from one opposite to the second column vector, the latter also wrote here. Coming opposed to the same core values. Now that we understand that they have been independent of each other linearly dependent Do We account for determinant. Because of a previous column, if this column is dependent la determinant of zero pulled. Here we see that the zero determinant of check accounts. No matter of three values here. The important thing is that there is zero. So here are three lines are three columns, not going out three for her, were three such chance. And we understand here that Q It is a non-singular matrix. Therefore, there is also opposite to the determinant is not zero. We also know the opposite. Already we like to diagonalization of this matrix We wrote: Q minus one we're writing. [BOŞ_SES] A'yl to do We stood and order is important, that you know the main theories. [BOŞ_SES] here in the summer of Q, I have to give it to us as a diagonal matrix lambda. Here was this danger: Two same value when it could be two independent vectors. Without independent vector zero determinant interests and it would not vice versa. Qin would not inverse. Reverse Q. Then we can not do this operation. But since, as we saw here this determinant is nonzero There inverse matrix, and this matrix can diagonalization. This is one of the main issues we know from the theory. If we look at this example, it is a matrix given as examples. The first line, second line, third line, type them in a row. A bit of a special matrix, especially, I give. This is solved in the same text. We see now that the core values. It's five core values, As in the previous example, and also he repeated twice. There is a problem in the five main problems now. Has calculated eigenvectors from five opponents, he found a zero-zero. Q writes in the first column. Was calculated from an opposite, he wrote. We found him. But, there is a second one more, he wrote that calculate the opposing him. Here we see the observed with the naked eye as well. See Cons of the third row and second row. And if you turn the negative positive, the same number, if you turn this minus old, same number. Thus two columns of this matrix is not independent. So it has to be matrix determinant of zero. To check this, just the event, in order to understand, We are going to ensure that the results of the summer. Here does the opposite for the Q determinant is zero. Behold could want here Qi = inf (Q) said, he would not give you the answer. And so here, although almost five an a listed state in the matrix As we saw our need because of a previous not diagonalizable Q we crossed with reverse. Right to strike again will Q'yl. And it's got to be a diagonal matrix. But the inverse is not Qin Qin reverse we can find here lambda, We can not diagonalizable. It is written on here but it's just something in terms of their representation. If it was not for Q inv determinant zero, hence this does not diagonalizable matrix. The same problem as in the case given in the main text should explain a bit more. Thus we come to the end of the examples we have solved on MATLAB. My advice to them with problems in the main text You can understand it better by comparing and contrasting it with the solution You can absorb more of this as well as usage in MATLAB. See you in another course in the future. Currently, linear algebra we finish our problems in the two volumes. First there's the second one you're watching, I'm doing this promotion. Goodbye.