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20Iv ' ' ;. ` [ _AzL21...

Question

20Iv ' ' ;. ` [ _AzL21

2 0 Iv ' ' ;. ` [ _ Az L2 1



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$$ \left[\begin{array}{rrr} 1 & 2 & 0 \\ -1 & 4 & -1 \\ 2 & -1 & 0 \end{array}\right] $$

This video, We're gonna go to the answer. A question of a 13 from Chapter nine White three for us to find the inverse off. The matrix minus two minus one 210 31 minus four. So let's combine this with the identity matrix once there is, There is there were once they were serious, they were What? Yeah. Reduce. So that's that three altitude of the first row to the bottom room. So that's going to go to zero. Ah, mine is 1/2 minus one ad for you, too, is 1/2. I want us to be over twos to you too. Keep a zero and one. And let's also add one of the first road to the second round. Get rid of this too. We're all scared of this wall. Uh, this becomes a wall 10 Top row stays the same minds to you. Minus one. Ah, whoa! 100 Next up, less subtract one of the middle row from Sapporo. So that's gonna be minus two minus one zero. Uh zero minus one zero. Minimo stays the same. 0011 What? Zero. That's also most black bottom are about to but zero minus 11 302 Next up, let's subtract one of the middle row from the bomber. That's gonna be zero minus one. That's zero at three months. Ones, too. They're minus one minus. Y T minus zero is too. Keep the middle. Where was it? Is seriously, Rabban. War hero. And keep the top roses minus two minus one. They were. They were minus one zero. Okay, let's subtract one of the bottom row from the top room. So I'll get rid of this month's one at zero minus two is minus two minus one minus 110 Uh, zero minus two is much too, kid made about the same there. Is there a woman? Rome on zero on dhe bomber. We can multiply by one, get zero. What? Zero. So most by my minus 1010 too, because minus two minus one equals one. T because minus two. Now, what we can do is most by the top row by minus, huh? It's gonna be born zero zero. Whoa, zero. What? Now you'll see that the form two rows. Uh, if we just flip these around, which you can do it, bro. production on. Then we get the I don't see a downside. So 010 That's what was the bottom room minus 21 minus two in the middle ground, which is not about tomorrow. 001 Well, zero on Dhe. This matrix here is the inverse off the matrix that we started with.

Okay for this one. We have a is equal to 12 negative. One 011 and zero. Negative 11 So the characteristic equation for this problem is given by negative Lambda Cubed plus three Lambda squared minus four. Lambda plus two is equal to zero. And when we solve this equation, it's a cubic equation. So you will get the three argon values as 11 plus I and one minus. I noticed that these talking visor complex congregants. So now if we have the Lambda equals one, then upon solving the system, eh? Minus I times u equals zero. We will get that use equal t times 100 and for Lambda equals one. Plus I, using a similar process, we end up getting that you sequel to a T times I'm minus two negative I and one no, for Lambda Contra Kit equals one minus I, which is given here. Then that implies that you is equal to it. Turns out that the Egan vectors are also conflicts congregates of each other. So you was simply gonna be equal to a T times negative. I'm minus two guy and one

In this video, we're gonna go through the answer to question number seven from chapter 9.5 West to find the argon values nine vectors off the matrix A given here. First. Do that. We need to find the determinant off the matrix. A minus. I ascetic with syrup. So this is the determinant. Off one minus R zero zero two three minus. Ah, one 0 to 4 months are You're wasting your over the top road Got one minus r times by the deterrent box Right to Buddy Matrix, which is remind us ah times by four minus R minus two. This is gonna be a one minus ah times by Ask what minus for our minus three hours. Minus seven are plus 12 minus to which is plus 10. Good. Then we can characterize this with one minus R and it's gonna factor Rise to be Ah. Minus five, uh, minus two. So we said that equal to zero. Then solve we're gonna have I can Values are is equal to one. I was equal to two on our secret five. There are three aiken values. Find their associates. I connect this starting first with I come back to rise. Because of what? When it's find the matrix a minus one times high times, the fact that you want is equal zero up. This is just 000 Good two to you. One on zero two three touched by yuan is equal to zero safe. Let's complete this by hand. So if you want, let's let the components be ex wives that then from the second World, we've got that to x close Thio. Why course Zed is equal to zero on from the third room we've got there, too. And two, Why close? Three is equal to zero. Okay, so this his bottom equation tells us that, uh, why is equal to minus three over two times that on the top equation tells us that thanks is equal to minus that minus half that minus why that's minus half said, plus three up to said. That's just ones that so, So X equals said, Let's just let them be ones. And that why's it was minus three or two time times said, which is just three. Everything. That's a first I Greta, the 2nd 1 we calculate with a minus, the less I connect it was too. Okay. Months to I times you want you could see. Right. That's gonna be a minus. 100 two 11 zero two four months. Two is two tells about you want zero? So the first hotels is not the first component. If you want equal to zero, then the bond to Rose. Tell us that Thea, the second and third components are next to each other. So if the 2nd 1 is one that the books reminds one, let's find the third Aiken Vector. Yeah, I can. Value was five comes I times you What do you want? It was there. So therefore months 400 two minus 21 02 minus one you want you see what Zephyr therefore you want? Well, the first component cto read off. That's just gonna be zero then. Second and for components are gonna be Will be, uh, yeah, we can tell from the second or the third World that if the second component is one, then the third component is twice second bone in, which is just too. That's a final act of Beckett

Ahead of their. So for this exercise we got this victory you and we need to calculate the projection of the victory you on the subspace. W. That in this case is the generated by these vectors we want and we do that are defined here and here. Okay. So we know that the projection of you on subspace can be calculated as the inner product of you with the generators of this uh Subspace. So in this case generators are W. one and W. two. So the formula to locate this projection is you want. They're probably going to be one. You The inner product of you would be one and everyone divided the square of the norm of the vector B. One and the same for the vector V. Two. Okay, so we have this formula but it's always good to have in mind. What is the geometric picture of what algae arrest one you. So just to give you some geometric insight of what is happening here, even though that you can observe that these subspace, this super space W. Is our four. But still this picture will help you two to visualize what is happening. So let's suppose that this is the plane generated by B. one and b. two. So here is B. One, Here is v. two. And these two vectors will generate this space. Stop the point is that you have here a vector U. And you need to calculate the orthogonal projection. That means that you from the tip of you, you draw a line or a phone call to the plane W. And then you want to you want to know what is the projection of you and me. That means that this vector you will obtain this factory, this factory in red response to the projection of you and the subspace. Go. What this formula here is saying is that if you can observed is the projection Of you in V one. And this part here is a projection on B two. This part here, this part here is a projection of you overview too. What this is saying is that you project you on the one. That means that you pick this this victory and only two. So you pick this vector here. Sorry, it is a different color on the two and you obtain this vector here. So this victory christmas projection of you V two and this in science corresponds to the projection of you the one if you some these two together, what you're going to obtain, well, you can observe is the projection of you on the subspace. W. So that is what is happening here, geometrically speaking and wealth. Just to give you some geometric insight that is always important to have in mind when you're mhm dealing with linear algebra, that will help you for future topics. Thanks. Okay, so this is a geometric picture and it works for any space. Now, we're going to calculate what we need. So the rest is just algebra. So let's calculate what the components that we need for to apply in this firm line. So let's start with the inner product. So the inner product of you would be one. It's going to be equal to two Plus 2. That is four. The interpreter of you would be too it's going to be three us Feinstein's two is 10 plus minus one Times -2, that's -2. And this is equal to Then we need to calculate the square of the norms of this two vectors B one and B two. So that is just remember that the square of the norm corresponds to the inner product of the factory with itself. So in this case they Square of the norm of b. one equals 2, 18. For me too, These equals to 36. Okay, so we have all the components. We just need to put all together on the formal obtain that the protection you on the subspace of U is equal to to ninth Times The Victory one. That is zero one -4, -1 plus 11/36 Times a vector three 51 under the final result after some these two vectors together is 1/12 trends, 11, 21 -7 and one. And this corresponds to the professional projection of the vector you in the subspace dog.


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