[BOŞ_SES] Hello. In an earlier session, linear place in algebra calculus, After what has sub-branches as we see it. We made a general tour of the horizon. Now we'll get what they teach here in the plane of the vectors. We will create an infrastructure. We will create a önsezg of. Yet in the subsequent section, the two unknowns, again when dealing with the solution of the equation seems simple, We will see that we have learned the very basics. The first three sections to create an infrastructure to develop an intuition made and can be generalized to the right where he gives us the clues. Then we will begin to linear spaces and linear section In space, the vector in two dimensions gidebiliriz of how we learn, we will see more general. And we will see in their Fourier series also followed by a linear function space. Fourier Series is actually a function space. A representation of space and the digital a mathematical method that most influenced our world. Digital televisions, digital dictation devices, communication, CDs, DVDs and the use thereof important applications. We will see processors after the linear space. So an alien to, and the relationships between elements in a space, the expansion of the concept will be a sense of function and after seeing these processors will also move to the matrix. So it called the operator of these processors. Number of series, how the two dimensional arrays of numbers We will see can be shown; and seeing operations with the matrix, now entirely ourselves matrix We will bring to the process and makes application. This calculation method will be made with square matrix. Because there are significant privileges of the square matrix. This constitutes the second part. So the second part of the course. After this course, the second course. Here we present a summary of Part 1. Because some people may not know Section 1; Some Sections 2 can know. These therefore be administered as two separate classes. Something important determinant of square matrices. Calculating the inverse of a matrix. Obtaining very special structure such as eigenvalues and eigenvectors. The introduction of diagonal structure of the matrix, but issues such as matrix functions defined in the square matrix. So you think we're all made up of what he saw. Now we're starting to Chapter 2. What we learned from the vector in the plane. Vectors in the plane of course everyone has experienced in some way. But to remove the essence of the job and not being able to draw so two sizes, the vector beyond the three dimensions, even infinite dimension function how it can be generalized to remove the space, there's a benefit to our core vision. There are three ways to identify vectors in general. One method of geometry, ie drawing. Something useful, but also to pass beyond the plane. Not being able to go beyond the three-dimensional space. But this is a useful approach to develop our intuition. So, we start. The same process, we see that we can do with the number of pairs. This is instead of the geometric methods, it leads us to pass to the algebraic method. Yet in the plane and in more than three dimensions, We will do the operation in space. They need to be able to pass further than a more abstract structures. This is possible with propositions or axioms or postulates the team. What the essence of the issue with this proposition enough to show us some simple assumptions that starts and so simplification is also very common to us paving the way for access to space. This will be the method of linear algebra. Vectors in the plane, should the geometric representation, as well as the algebraic representation this will become fully generic simple applications of linear algebra. But we're starting another ordinary. Because historical Gelişli be the case. We do not get this intuition, very abstract and the beginning Since we encountered a situation that is difficult to grasp. However, although it seems as simple, vectors in the plane that will give us an extremely important preparation. There we go. Let's start with a point in a plane. We call point. At this point, we need to be able to determine the location of the two coordinate axes. When we combine this point in the beginning a determined by the direction and length of X we are got. We call the vector. We call vector in the plane. Is it in two longitudinal lines of this vector, ie, absolute worth a little more different than he is sometimes shown as a single line. The illustrated embodiment shows the vector length. What if you are measuring when you see him here. There are also a unit vector in this direction. This unit vector, the same vector obtained when the longitudinal direction of the vector interrupt. As long as this is already a description. Unsatisfied with any vector can show the size and direction of the vector. So let's divide and multiply the vector of the longitudinal length, shows that parts of the unit vector. Here we see that the size. Now of course we can not do anything more than a single point. Let's continue the colon. A point A and a point B and get them Let us define a vector combining the EU. Again these vectors have a length of a de direction, have the direction. B. We define the vector going from the EU. He is going to define the vector from A to B as B. Now we think that we have a lot of vectors and points like this, a tidy them to do the same batch job We have to start from the coordinates of the center. The way that the EU can move parallel to itself by being the city center, The point; We see here at the center of the EU vectors. Similarly, the B vector at the center facing B. We can move and size of these two vectors as you can see, although the same, We see the opposite direction is a one. And the B vector, opposite vector of the EU that happens spontaneously emerged. Now let us assume that a set of points instead of two points. Let's start with the three dots on it for the first time. A point to point B and point C, we can ask a question like this. I want to go from A to C. Using this given point. How can I do? I can go directly from A to C. a. B. Using the three points from a well, Go from A to B, from B can go to C as well. Now that we have defined on the previous page let's go through vectors such as moving to the starting point. EU on the future of the EU at the starting point of this vector It is shifted in parallel to it. BC is the starting point coordinate team the beginning of the future, Shift the parallel to him again. Shift the similar as the AC is parallel to it. Here we encounter a very interesting situation with itself. EU and parallel consisting BC we think it's going edged rectangular AC diagonal. This occurred spontaneously. We get it, we force necessarily say that this is already an important vector algebra will be a success. We are here to say that EU AC When we add the BC we arrive at AC. Here we come to AC when we added the EU BC. So we can define it as a collection. AB + BC = AC, where we define the way through wherein the sum of the diagonal of this parallelogram diagnosis, we are identified. Parallelogram rule emerges spontaneously in this way. And thus it is beneficial for the natural self becomes a rule. We start with two jobs in vector algebra. One collection. Recently we have collected in BC with the EU. They are a bit more abstraction for the EU to x, y bc say. X. way to y collection, where x on y we obtained bring in, to reach the point directly to the C x + y = z We call; or parallel to the y X. We're going to have collected the diagonal edges. So we are equivalent to each other have found two collection rule. But the gist of the rules we define as a parallel sided. When we get the y obtained on this x It emerges as a sum of vectors in a natural way. However, when a vector x is given, thereof, If we triple the received as in this example, three times in the same direction that is meant to enlarge We define and accept it as a rule. As we said before we We began: the merging point B to point vectors have achieved the EU. To the point, the vector that when we combine the starting XI to say, B is also a vector that we get when we connect start XB say. Again the beginning, AA, B. Suppose we come from there using the EU. So here, we're adding to the EU on XA. XB is going on here we found itself in size. Here we had the right xA, We have found the EU is expressed in the vector of this point. Assuming the contrary, we find how he bade, We arrive at the beginning of the XB see on per attaching point. That means that if we write the equation A + B = X to XB, here xB We can find the time we spend per side. BA da XA- XB. We arrive here with the collection rules to ensure consistent results. It says we're having a kazanımı: EU vector one by one, they are finding the vector discovered the path of these points. Alone, that the need to pay attention. Important here as well. At first intuition might say that the EU, we also vector minus the vector B can be. But on the contrary, the second vector, we are removing the vector of the first. Vector of the second A, X, we would remove the xB first vector. Stressing that it was important in this sequence frugal with parallelogram rule, compatible, let us continue our progress, saying that they are consistent. Some examples of how daily life is going on in this way? Again, you have points B and C space. In this, we sabitleyel them A and B location. I also necessary to introduce between these two and a rope. and somewhere between the two ropes, in a weight C prime. Now how these forces are distributed in the rope. This is a major problem. It is found, the work of the vector X AB AC cord it carries the force of BCR y x carried by dint of rope This means the balance of the total force hanging down. This time we take the reverse hanging down force, demonstrate that it is equal to x and y, the total force of the hanging. Yet still a daily practice, a concrete application could display. The inside of a river or a canal In a boat, the force in the x direction, Suppose we take the force y in the other direction. This offset will move in one direction between them, of course. Question may be: how the x and y forces My pick is advanced parallel to the canal boat? The sum of these two forces pulling M kayak. This means that the sum of x + y should be chosen so that it is parallel to the shore. This embodiment again. It is an application of the two parallelogram rule collection. Let's take a further embodiment, a broadcast end, a unit weight primes. This publication height, will extend so far. Twice on this publication If we hang a weight, length will be extended two times. An embodiment of obtaining a doubling of the force. Yet they still get concrete examples although simple, instead of taking a car for a horse, two horses and forces equal to twice the jump move those forward force that takes this car, It will be the value of a force at both times. So i will be a force to the one embodiment of the shock. We have worked with so far way. Of course you can see them in a physics course many Or, apply a technology, engineering courses. We have now been here, we limit ourselves to these concrete examples. But we want to proceed. Our progress is there, in good shape geometry tells us what's going on. However, it is difficult to work with geometry. You will draw the neck. It's not always easy. The job will be much easier and you can put into But figures It will open the way to progress. Now we, of any vector x, x1 and x2 coordinates the team again, the projection on the x1, y direction Let's projection of x1, respectively, and their length and the length of Y1 say. If you already know us by this Pythagorean theorem that the x1 and x2 We know the length of the vector. And that define vectors. Let them as a unit vector. x the vector direction of the unit vector i get. Of course, because it is a projected zero in the y direction, a call to the component in the direction x. j is the opposite. Complementary. It does not have a component in the x direction. y direction of a component. So here we strive to exceed the number of vectors. Indeed, we know of computing a vector of a solid. i vectors of x1 we take the floor, we get the following vector. vector in the direction x1, x2 we take the multiplication of vectors j, We learned this product also geometrically. We take this vector. The sum of these two vectors, that is the initial vector x, It is consistent with and vector collection of certain rules take a strict line with the rules, we have seen consistent. That means here, it seems possible to provide an accurate advance from geometry to algebra. If we continue this progress we still vector x, We show it with x1 and x2. y vector, we show the Y1 and Y2. Here, here's move parallel to the y we get. Or, in the triangle formed by the large y, Let the length of the edge on the x-axis of this edge. This provides a simple geometric triangles through simulation, that is 1 x 1 plus y z 1, you can see that it equally. So this year when you add 1 to bring the end of the x 1 You provide this equivalence. Similarly, you can do the other side. So z equals x plus y to arrive, these components x 1, x 2j y in y 1, y 2j have collected these edges obtained by the x 1, When you add y 1 x 2 y 2 when you have time you add, You will find that you have found it equal to the same vector. At the latter easier to see if there is a vector x x 1 with x-axis projection in the direction of If you get this car if you take the floor three times, for example, here you see each other two homologous triangle. If each of these hypotenuse c floor, edge on the x-axis, is also the same rate. Similarly, we obtain the complements thereof wherein The ratio of length of the triangle, the length of the hypotenuse income ratio equal to the ratio of length on the other side. So here we are also continuing our efforts again in geometry algebra we find the number of results that is consistent with, isomorfizm it called as a term that made international peers. You know iso mean the same, equal isobars means such as pressure, morphology form of polymorphism means, likewise in the same algebra and geometry shows that the work we do with this definition. We show here i and j with a x and y are vectors of vectors on the previous page, but we see that, i and j are also able to demonstrate the use. This is more economical, affordable way a spell, a simpler notation. As we bring a definition for it; i and j are the use of a vector x x 1 x 2 and y is the vector y with the number 1, we show the number of years 2 the total sum thereof corresponding number of opposite, first Among their numbers, the second number we get by collecting among themselves. We say that x 1 and x 2, the components of the vector x. I towards the one in the x direction, j direction or the other in the direction of second year component and the aggregation rules gene We see that it is compatible with geometry. Gene X. xi means to multiply the number of C, components We have seen that the individual is meant to hit the same number in the previous figure. x if we wanted to find a tall, The length x along the horizontal axis x 1, along the longitudinal axis x 2 that is an equilateral, We're sorry we get right triangle. x 1 in the direction x 2 in this direction. We know from the Pythagorean theorem that, The length of the hypotenuse is the square root of the sum of the squares of the other sides of the neck. So, We have reached this definition is also compatible with the Pythagorean teoreim results. Yet again we want to find the unit vectors of these components, x 1 and x 2 components controlling stake after the paint. Our continuous effort and as you can see, vector of the number of drawings that provide the pass. Quite a few also, we are saved from this stage. Let's do an example. Recently we talked about daily life so that a sample board over from a center, Jackal a nail -1. A jackal nails 2. Belirleyelim a point in three points along the vertical axis and combine them with a rope, a weight I hang him on the spot. The coordinates of the point across to digitizing x three zero but minus the y direction, the x-axis such, the vertical y-axis the time and we accept here We put a block of six units. Where B is found -1, C is 2. Actual force, wherein [BOŞ_SES] We want to balance the forces in the rope. This does the opposite: Downward -x, has harsh force, x plus y force minus the force of our forces will be in balance. This will be zero or z must be the sum of x and y, -z'n's, hanging down at opposite forces. And it's weight, The real weight to the point that we want to find out how to disperse the rope. Now we just rope the EU in this direction a force so we can find pulls the direction of the force. Direction is EU. A and B are given coordinates. So, the EU vector x B vector, We found by subtracting the vector x. Similarly, the AC line to clear. This AC vector X by removing the gene that the vector sequence XC important, We extract the coordinates from the first second, we find it. This means that the vector x is the direction the EU but do not know the length. One EU is the x solid. We get it. We found the EU. We bumped it with x. X unknown, unknown year similar. But our known size. When we brought them together in the z We're getting out of nine weight numbers are so beautiful. Of course you can get the desired weight. x plus y will give a total of z's minus x plus y or z is zero to compensate. Let's write in terms of these components. x vector x times the EU. here we find the vector y y (2,3) z the vector of a given size, there's the unknown. Zero nine times, nine in the weight, weight hanging down. We plus 2y is equal to zero -x when we organize them. 3x plus nine will also 3y. Two unknowns, two equations. When we solve for x and y, here x two, y, we find one. Indeed if we look at the supply into place, x instead of two; minus two years instead of one, two, giving zero. x instead of two, six goes on, rather than a year, are three. A total of nine still able to provide this equation. If we place these which x and y are in place, We find that x is in the EU rope force, this force is achieved by two co xi. The Y force accumulated in the AC power cord, power distributed there. We found him going. From here, the lengths of these vectors If we see that we want to find the amount of force we take the length of this vector. The first force plus two squared gold rope forces in the EU We see that 40 is the square root. The strength in the AC cord y [BOŞ_SES] we find that long. So two squared plus three squared, which turns the square root of 13. So the AC power cord rope up and down in the EU We see that a force scattered around twice. This instinct also, consistent with our intuition. where the EU is doing with vertical angle more acute angle, it will carry greater force. This is less to carry less force than if the rope is tilted. It is consistent in this way, we provide is consistent with your intuition. Now we continue to pass algebra to geometry, ie Our goal is to make getting away with a number of works of the way, come. In space and Let us unite these centers and B points. Here we create a triangle. On one side of this triangle vector x, with the other side intermediate vector y, and let's get the theta angle. With this geometry, algebra again to reconcile OA progresses we know that x is long. Type the name of the vector length of the right and left, We define with double vertical lines. OB is the y, we see that the length of the vector y. We see that the EU is the EU vector x minus y. Because we find the attaching y x EU. So yx plus the EU, so the EU is equal to x minus y. This aspect also gives the length of the EU. Now, we use information from geometry. The length of the sides of a triangle other The length of the sides and this formula is given depending on the angle. Pythagorean theorem as a special case of it anyway. Cosine theta zero so that the Pythagorean Theorem is a right triangle. Tete ninety degrees which is the hypotenuse AB. Hypotenuse squared, the square of one side, the other side of the square angle of ninety very least, the product of two times the cosine of theta edge coming times. Now in this geometry language Let's write it in terms of language or vector algebra. EU y minus x minus y, but the same thing as x height x minus y size. So let's write that x minus y squared neck. O x squared, x the length of the frame, OB and minus twice the square neck y O x of length, OBE this year, including the length and cosine theta vector equation with algebraic geometry It is expressed has translated into an algebraic equation. We see it as a group we do not know where we are here again this summer. Because the neck is something we know what the length of the vector. Length of a vector is a vector of length again is going on here is a new group formed. This will be important and in other places. We call it a defining for him. We say that x times y times the height times the cosine of t let a new name, this point in x and y show, let it be domestic product. So x and y vector inner product of the vector x with y. This confronts us expedite the definition. Now I wonder how we can do in terms of this definition lengths. See here minus two XY Let the left, Let x minus y right at the square neck. We get the following equation. Because cosine of theta times where xy x y point becomes the product. So this equation completely re-using the inner product definition there I'm typing. We collect unknown on the left side. The reason for this is because this writing. We also collect other vectors known to the right. How is it twice the length of the inner product x squared. The sum of the squares of the components. y is the length of the frame components, The sum of the y component of the square, where there is x minus y vector. x minus y vector x component, y component. x component y component of the first and second components. One edge of these is, that the frame, the other edge of this frame. We are experiencing a very nice surprise. There are a lot of simplification. See here, x1 squared, when we opened this square braces x1 squared in the square comes from biirnci comes y1 squared. Both of them are here. But they come with axes, here too the increase. That means they are taking each other. Let's open a second term in a similar manner. X2 Y2 plus the square of the square. See X2, Y2 are the square frame, which in turn leads each other. But there are other terms. There's one x1 y1 minus 2 times in one term future here. Minus 2 times X2, Y2 also has a negative start, plus a minus because it comes from here Two factors common multiplier factor of two partners x1 x2 x2 y 2 taken by the pros come out. Now that the initial interim forgetting details 2 times the sum of the product with internal cross to the right algebraic If we compare, we started with a geometry of the inner product. Let's have a new name to the product of length cosine theta. What we are going in terms of its components that have the algebraic calculation, we're proving it. A proof of this theorem, but the inner product in a simple geometric theorem also acceptable definition. We were also able to demonstrate the voracious start here. Yet here whereby X. y domestic product, tetal was giving the cosine terms of the length of x and y, we know. So the cosine of theta divided by the size of the domestic product. We found out what the inner product. The results here in this denominator, the denominator teoremiyle pisagor in the first round, the second round teoremiyle pisagor. We are winning viewed geometric coordinates a size we are able to express in terms of the strength of the won. This is an important advance. We can ask the question: Actually, cosine theta we have found but, When the sine of theta cosine of theta there is also. Sine of theta cosine of theta, of course, can be taken as 1 minus the square, but I wonder sine Is there a process that identifies theta also so we can ask directly. The answer is yes, but we'll do it here. The following terms; something you would see this as a vector multiplication of physics necessarily. But you will not see here does not matter. Because this product useful in two and three dimensions. More sizes can be defined a little forced, but internal Not so handy product. He respects here we limit ourselves and wonder cosine tetayl who earlier this Korser to it as a vector multiplication They find it very variable functions in the sequence given in. Now let's try to find the properties of the inner product. Domestic product, meaning that we'll see come to a projection. Refer to the following; Let the same point again. Starting from points a and b. a projector remote OA OB I convey that to the other. We found the shadow of it. Here we call it, we call projection or projection. Similarly, the OB gene in an upright projector that time can call the OA's shadow. On this OB. It is available; We say we call OA or OB projected onto projection. Let's get them from simple geometry, it Find algebraic equivalent, and so we will find out the meaning. The inner product. He's one of those here, We create a right triangle with B and H point. This OB right triangle hypotenuse, We found that we hit OH tetayl the cosine. So OH, OB is multiplied by the cosine tetayl. We call the projection of x on y. on vector x, y is the projection vector. You see it in a remote projector, OB as the shadows. Similarly wherein x is We find the projections on the year. This gives us the cosine of theta is multiplied by y length. Where x on y algebraic equivalent of izdüşümün we find the following: OH slightly one can not but welcome the ear, H height of the international initials I never use; of y x projection on the OH. This is multiplied by the cosine y size. But we know what the cosine, x and y domestic product divided by x height, The length of y; there's also the length of y, but we see now that y height, simplifying each other; and the inner product, x, we find that the longitudinally split. Similar projections also, we find the projection of x on y. This is completely symmetrical to it, again dividing the length of the vector inner product, but this time the y longitudinal division. No need to memorize it because it arises spontaneously. Because of where the axis x we get; x of its longitudinal division, The unit vector; y is a unit vector v of its longitudinal division. See here then the first projection He once stood; a second projection at any time, there was v y y times x multiplied with the longitudinal division. Therefore, keeping in mind easy to say, The other type is not possible, y we are looking for the projection, We bumped it with the size of the place made projection unit. Similarly, we are looking for the x projection, We stood with the other aspects of the unit vector. So domestic product projection shows a projection. If we try to go a little further, We can think of two extremes. The x and y vector may be identical or each other It may be parallel, in the similar direction thereof; or X. y can each other. One parallel and equal, or with equal length but not parallel or perpendicular to each other. When the gene x with x equal The angle is zero, it means that theta = 0. theta = 0 when a cosine zero. So here x points x; y because we get x; x1 x2 squared squared plus will; This would be the square of the height in x, the Pythagorean theorem, we know that; some prefer to call it the ancient Greek Pythagorean; we provide. An edge component length of the x-directional plus the length of the frame's edge in the y direction, hypotenuse squared. The orthogonal y and x, then theta is 90 degrees, as pi radians / two; Cosine of ninety degrees zero, It means that x dot y is zero. If we write in terms of components, As you can see the sum of the product of the component is 0. The sum of the components we read that TERIS, the sum of the product If 0, It means that these two vectors are perpendicular to each other. According to 0 degrees in line with tet vector, a vector is a certain multiple of the other. This example y, where x is expressed as a certain solid We can; equivalent, it is obviously a solid y x. Specifically, we look at the unit vector, ie, (1, 0) through; j (0, 1) consisted den. If we look at the size of this vector, one squared plus zero squared, the square root of a length in the same way. Taking the inner product of these, 1- 0 Multiplying zero; JOIN with 0 to 1 multiplied by zero; so i and j at each other. Already these characteristics, this i and j are doing important. If we want to improve a little bit more of our review, What is the importance of vertical and parallel might ask a question. This question is a bit of mathematics, where the approach worked. Application in the x and y events, an effect arises as a force; in physics, in sociology, in olsaılık, in many places. If x and y are perpendicular to each other of these events, no projection on one of the other means, So what happens if an event means that the other is an independent event, means not affect each other. Or, if this is the opposite of x and y parallel, one solid and the other, means that events; events in which the x and y show, forces, an electric field magnitude, such a heat flow, it means the same quality and it is important to understand this extraordinary. There are two events, they did interfering with each other, it does not interfere? Or is this the same quality? Important to understand that. This immediately brings a strategy, an x and y to compare If you want; both given to you, always in the direction y x a component and can separate a component in the direction perpendicular to x. This is the year of the event, the event in which the vector y How much of the same quality with x, how to fold at an independent I understand that the event is the most important aspect of understanding the issues. They also make it as a metaphor, ancient Roman emperor Jules Sezarın method; all fetihlerinde Divide into small tribes that individual large countries Tribes take management; it's called divide and conquer method. It made little something like mathematics but also in a more positive sense. Allocate a magnitude component x with the same quality and with different attributes; that split in one place, Disassemble and thus can better understand, You can control better event this year said. Returning again i and j, j i and ii above no components in this context, No component of the i j; means that affect each other, vectors independent of each other and also important for it anyway. Now here's one, two simple examples We want to make more concrete issues. You are given an x vector component, as shown (2, one); y vector components (1, 3). We want to find their size, their We want to find the angle and trying to find the projection on the other. Although we've seen examples of this concept will allow simple application. Here u and r vectors. These vectors length easily. x 2 square of the length plus 1 frame. the length of the y gene of these components The sum of the square and the square root of 1 + 9. So here's painting, As we found in a previous page, root root 5 and 10. If we wanted to find this inner product, As we find the inner product of x with multiplied by the y component. If we divide them after we find cosine theta is the x and y length. This means that the angle between x and y, are able to cross to get to the interior. the inner product of x and y is between 2 and 1 multiplication, first component; and collect the product of the second component. And x and y are longitudinally split in them when we cosine We find theta. As you can see here due to the denominator 5 selected numbers turns out, Under the square root square root of 5 and 10 to 50 turns. If you get 5 under the square root as the square root of 25, half the square root of interests. Dividing the square root of one of the cosine or if you hit the square root of two, square root of two divided by two is 45 degrees. 45 degree angle in this way, as indeed we find here, visually manifested here. The last question on the projection of x y y was also projected on x. y x height x projection onto the cosine of theta is time. So here x height has roots on the root of the five events. This root five Cosine If you hit tetayl root projection of five divided by two turns. We can find it from geometry. Similarly we can do y. y x y is the length of the projection onto the root bifurcation ie two split root cosine multiplying teta'yl is doubled. This allows us to divide a root y size of ten times we find the root of two. We can find the geometry. But without something more to draw any geometry a costly business, precise, accurate to draw a more difficult task. Here now are able to get from the inner product. x projection x lengths multiplied by the unit of the y direction. So remember, to understand in terms of ease of mixing of memorization. Because we take the x projection. So x will be multiplied with anything. What will be multiplied? the unit vector in the y direction. The unit vector of the following important aspects: a vector in a milimeter also if you get a projection on a vector of one kilometer length The projection will be falling over if you get the same thing, you need to go. His unit length to come here. Similarly, we receive the projection on the y x so y We have achieved getting neck with inner product. Multiplying the y x unit vector. A second example: the vector x, let us again like the previous one. The vector y is not given completely given component, combined with other alpha unclear. And the question is: how we choose which of these two vectors alpha to each other You may be parallel with or perpendicular to each other? We also can handle with internal cross to again. Namely, x and y multiplication, If the inner product of these two vectors are perpendicular to each other it gives zero, x and y. So we find the alpha here. Geometric vector as the vector x given year, but only the first component is given a. The second component is an alpha hence these vectors and the alpha vectors form a family. A first component, second component, the residual alpha, As you change the alpha family is achieved by obtaining a new vector. Among these, as you can see there and where y is the vector base vectors shown in y with y vectors of the base opposite each other. Really are the vector perpendicular to the x. From here we can find y. This Referring more generally may be a vector y t times the Located. So all of the normal vector x can be obtained here. For the vectors may be parallel This vector x with y vector must be a certain multiple of each other. We do not know how many times it would be solid yet, but k and y vector comprising an alpha, We do not know the XK, we need to have a certain times of the vector x. See here by the first component in the left side, the right side two k. Here's a split that would remove two of the k. And the right side component vector k Multiply that by the time we hit the second component of the short time we got to alpha. Or, conversely here alpha alpha k is equal to k We find that a split is two. We can also call it the overall solution. Vector parallel here because there are infinite in length, but each at one time we also get a first component off vector. Now we continue to understand the vectors in the plane. There are two inequality. These are actually things we know from geometry. One cosine theta is smaller than that, cosine theta absolute value smaller than that of generalization. Now you can say that what we're winning? Here we do not have much of a gain, but in terms of vectors in the plane n-dimensional vector functions, and even the future last time that we move to slash infinite dimensional vector When these inequalities will become an important field of application. We know that x points x and y so y interior The one aspect of the product multiplied by the cosine of theta length, taken in absolute value cosine theta is coming just because the absolute value of these tribes are already plus valuable. Cosine theta cosine of theta it because it is smaller than sheep instead of something greater than the absolute point x y that is greater than the value of the product of this size, We find that little sorry. On the other side of the plane known again You get something a vector y is a vector x. This happens when we combine the two x plus y vector. We know that the length of the vector x and the vector y When we get x plus y sum will be greater than the height. So x plus y is the sum of the length We have to be greater than x plus y. Now let's write the following inequality. x plus y squared x plus y x plus y we find the inner product. We found earlier. Length plus the length of the second hit of the first product of two vectors See the cosine of the angle in between, but x plus y plus x Pythagorean theorem so that the angle between the y zero. For what? x plus y vector is applied to the Pythagorean theorem. If we turn right side, the length of x squared plus y is the length of the inner product with the x plus y occurs twice. But we have just seen, the x and y inside multiplied because of the Schwarz inequality it also shows that the cosine of theta is smaller than one. There is this inequality. If we place these inequalities x plus y that x plus y we put into something bigger, We locate the merger cosine theta. [NOISE] This equation turns inequality. And something interesting is happening on the right side, it turns a full frame. x plus y is the length of the length of the frame, All the time we take the square root of x plus y length, Length of length x plus y that is smaller than the total of the neck fracture line indicate that taller than a straight line. As I said to them very familiar things again plane The obtained also indicate that once consistent and also the future of our great drawing of three-dimensional vectors in a manner that is not possible Or, the function will prove to be significant disparities in space. We here in this infrastructure. Perhaps you could say something known known but more abstract Moving on to them more meaningful and linear space inequalities that are beneficial use. Here we repeat these things here. The length x height x plus y length plus the length of x plus y is greater than y neck. The length of the straight line is smaller than the length of the broken line. This is a broken line, the straight line x plus The length y is smaller than the length of the broken line. This is something known but from three large space and infinite space in which this generalization We understood here as a more intuitive way to that result. Here this x and y vectors vectors again we selected earlier. Schwarz inequality and the triangle inequalities in these We want you to provide. E x height particular. I do not want to spend time on it too long. The first component of the second component plus the square root squared off five. The length of the first squared plus y squared root them comes second. X and y the time we collect turns three and four components. Two, one, one, when we collect all three. We calculate the length of time that the three plus nine frame 16, 25, the length thereof makes five five. Means to calculate the inner product of these components fit well the first of the first, then hit the second to collect the second. Multiplying one with that I, the product of the merger of three gathered turns five. Now Schwarz inequality was as follows: inner product of this size, It will be smaller than the product. We found the inner product of five. The first length, the second length. 50 When we crossed them off the root. He had seven out of 49. It's like a seven point. Indeed, larger than a seven point five. Yet the length of the broken line x plus y, sorry, The length of the straight line, the length of x plus y here the length of x plus y squared plus four squared, we find three, five. This feeder from five different backgrounds. Root root of five plus ten. Now we calculate it a little, a bit older than these two. 10 is also slightly larger than the square root of three at 10. Thus we see here is a number greater than two out of five. Now here it is no longer in the plane We want to continue to work in three-dimensional space of the work. It would do a break for it. This is what we have learned to thoroughly revise our sindirebilme.