[BOŞ_SES] Hello. We saw our equations of the previous session. Our headline simple equation here. This is a very special type of equation. So the right side of the equation 0. If we interpret this in terms of conversions it means to find the null space of this conversion. What we saw earlier in the null space. Surely there are 0. But I wonder if the problem is that there is this one other than zero vectors as well. Turkish These kind of simple equation equation even call the homogeneous equation. In this equation, x = 0 only There is one ÇÖZÜMSAN it 'empty' solution or Turkish in the 'trivial' it said solution. We generally like the only solution. Because that's certain, but in the future we will see some issues This simple equation will play an important role We want to get beyond empty solution and solutions. The solution can not be called to check on the Turkish non-trivial solution. We preferred it blank solution. This is not something new. If you have only the zero vector space in zero Do you have some vector outside the zero vector? There's a bit of a short section to highlight it. Let's make it two concrete examples. Our first transformation T (x1, x2, x3) still brings in three-dimensional space and their answers 0. So if we separate them into components, x1 + x2 = 0, x2 + x3 = 0 ve x1 + x3 = 0. There are three equations for three unknown. Now, of course, x1, x2, If we see that each place x3 to 0 in this equation we provide. That means that there is always an empty solution. Issue of whether the solutions out of this solution. Here we see easily. Subtracting the second equation from the first equation x2 disappear. X1- x1 and x3 = 0 = x3 from clear that these two equations. You find another vision x1, x2 also attaching to 0. You find also attaching 0 to x2 x3. This shows that it is equal to x3 x1'l. X3 = x1 we came here we use means that two x1 = x1 = 0 and one has to be 0. If we understood that x1 x2 0 = 0. x1 x3 was equally well, he can 0. So this equation is an equivalent to this transformation by adding a right you get. The only solution to this equation is blank solution. This may sound interesting, but not interesting solution is the only solution to be empty. We will see in the future especially in the eigenvalues of matrix problem Or, eigenvalue problems are also called eigen value also called on Turkish as well. There's not a solution that works, we will see an empty solution. We will seek solutions that are not empty. Just a blank sample with no solution in the next example. See a similar transformation to a previous here. Here again the first component is 0, The second component of 0 and 0 if the third component, now we get three equations with three unknowns. But here we see that the first two equations are not independent from the third equation. Summing up the first two equations x1 + 2x2 + x3 2 would be. It gives this equation. Therefore, even if we have our three equations as two equations There have independently. So we have only two against our three unknown equation. We can not use the third equation. Because it carries information that is not different information than the other two. Therefore, we say T to one of the three unknown for example, we find that we also x3'et the X2 X3 -t. x2 -t ise x1 de t olur. We see that here means that a non-null solution. There are infinitely many solutions, but all of the infinite as we have seen before While reviewing this zero space (1, -1, 1) of vector layers. That means there is an infinite-dimensional. There is also a non-null solution of this equation. He does not teach you much emphasis something new. But the emphasis for the future will face a very important kind of equation I chose. We end this lesson with this section, that we no longer bet. Here are some assignments. This linear space, the processor definition, the target space, the space shuttle and space determine the space and the space that is 0 for each Determine the team said base. We have examples like that. From finding the solution of the following linear equation, it is important that, because two very simple equation equation with two unknowns. Solution, whether it is that we have seen in bets If you think we are removing from the order of the conversion. null space is empty so you have only the zero vector, Because we were enjoying Is there another vector. We are removing is in the right side of it is accessible space. So in order to reinforce these concepts that we expect you to do the homework questions. Look for these is still the linear transformation. Make some operations with them. Also specify whether they are defined. For example, if two definitions make an impact on the conversion of two-dimensional space The two-dimensional space defined process that you would not if you give a three-dimensional vector. Yet, as we always do at the end of a chapter and tidy passing through here that we give summary So you can see the issues that we did a combination of both of these short You understand and the less you understand presentation To reveal where a car easily, I think. I myself in the later half and still a student, I do it in the course of time will tell. Where would you get the details with emphasis bird's-eye view of this summary helps a lot to understand the issues, I believe the plunge. My personal experience is always the case. I pass it quickly here. Here are four space definition when defining the relationship between these two space needed space, zero space as the target space and transportation space. Here, we talk about a very important theorem between these dimensions. All this gives you a chance to test so I hope you understand. That has to be one of the solutions of equations here Or, that solution is not the solution I offer a summary of the case. The next topic is the linear processor and will pass the matrix equation. How to draw the vectors are able to do something, but we go much further, not possible. When we put the number of vectors also can not draw it So the two dimensions and three dimensions in a higher dimensional space, even forever also it enables us to do things in dimensional space. So this transformation to show the number of linear processor We will see the road. We are taking them to the matrix.