(b) /ipt] find the determinate of the linear transformation...

Need some help with this Linear algebra question really confused by it

Here in discussion, we are going to calculate that transformation metrics that Rotates The Excess X three. So here as we have given that the excess X three of are rectangular coordinate system is rotated bye 45 degree. The words Excellent. Around the X two access. So to understand the situation, we must draw uh figure first. Now here in the given figure access. X men desh and X three dash Arli India. Excellent. Extra plane. So from that given figure we go Excellent dash equal to Excellent. multiplied by cause 45 degree minus X three multiply by of course 45 Diggory. Therefore from about we get excellent dash equals two. Excellent divided by the square root of two minus X three divided by the square root of two. Yeah, X two dash is equal stew extra and X three dash equals two extreme where people are by Close to 45° plus X one multiplied by because 45 degree. So from a bow we get X 3- Equal Stuhn excellent divided by Square root of two plus, extremely divided by the square root of two. Hence the transformation metrics is given by first and pastoral. It will be one divided by the square root of two zero And -1 divided by a scribe tough to. Now comes to second row. It will be zero one zero and comes to 3rd row of these metrics. We have one divided by the square root of two zero, one divided by squared off. Who. So this is the required transformation metrics

Students today I'm going to be answering um difficult question that was given to me. Mm So um the question raised and that is a true or false question. Um breathing the equation X equals X. Two times you close. Extra teams be with X. Two and X. Tree being free. And quotations if I mean in parentheses they were saying that neither you nor fee is a multiple of the other describe a plane through the origin and and and I'm seeing that it's a linear equation whereas have some difficulties of understanding. So looking into this um I have one way to explain the problem. Um The best to my ability. Um So I'll just use an example. Um I just use one simple example too. Make sure we understand um The question at hand. So I'll just use one one equation that we can use to determine what it is true or not. Which is 10 X one minus 362 Mine too. What? 63 zero. Uh huh. I'm the way how we might have to go about this. We might go about it where um where we might end up the fighting 10 on both sides where we know this will let's go down. Yeah that depends will cancel out. And you could expect hello? Yes. Um It's here right? Do you square? Oh not uh Sarah point to uh uh Thanks to uh which was both zero and four times think since we're trying to get it to um equal to um equal to X. One. Because um so uh X one is um it's like a free value. We can move mhm negative 0.3 and negative 0.2 on the other side where you make it X one equals um 0.3 X square um Plus 0.21 X three mm. Yeah so and then they also brought us in um and one the question that um that's two and X three are free. So the way I will create um the general solution of the um in terms of the factors um we'll set it up like I'm not sure if you are familiar with this setup that in these brackets um you have X one next to X three. Uh Just make it a little bit yeah better a little bit more neat and drawn out of it like that X. Three. I will have it equal to um Where for this setup? We will have um carol point three X square plus 2.2 X. Okay let's make sure we get a lot of room. Sorry I'm gonna make sure there is a lot of room. Okay To share 2.32 X three. And so that's so I set this up because that's what we got for X. One. Um As I'll show you up here because that's what X. One is. Um this top part. And now we got to figure out what it now we have to figure out for. Um Well being it's too and X. Three. So I'll add the equal where we're gonna solve one. Yeah. Hold on soft one. For um 0.3 X. Square plus. Um Hold on I don't like how my blood sign is looking plus um this bracket that has uh sarah oh my two X. Three. So the way how we will solve this uh So the way how you go about this in terms of solving for the free range um says with X. Two and this X. Two says the 03 X. They both kind of have similar people are kind of similar. We might see X. Two here and with this one it will just be Cyril because we're showing where um that placement is. And then for this step, the setup since you see with the X three, I'm sorry for crane. More confusion in this extremely it will be a zero here in an extra three. Um on that side my counsel making our general solution for the factors. So dan when you write it out um we end up bringing it out you'll get um X equals X square live sarah 0.3 one serial plus your X three which would be hold on. Let me make sure it's big enough where you all can. See. I'm sorry for parents not with this. Yeah. Oh yeah 3.2 and pharaoh one. So that's how you would do your set. So so based off with those solution we can set the solution is um a span. Mm Okay. And with the span it's just basically um right you mm see yes okay sorry this looks atrocious. So since now we kind of have a general idea of how to solve this. Um so determine whether it's true or not. We can try to draw no way you can do it, go about to try and draw like a plane. So here's like the graph. So this is like for um this one's for X one and this one starts X two and we can bring this out that will represent our X three and how we might go about it is where you might draw this plane going like this and draw another time going like that where this would be our fee and this would be are you? And the question did ask what their um at the plane the size of a plane through the origin. And we look at this plane is luke? Is this playing um going through their origin? You see that it is going through on the origin. So based off of that set up and the calculations that you see um up here on how to set it up. Um The answer would be true. I hope I explained it to the best of my abilities uh If this helps you out, I hope this helps you out. And I hope that that as some clarity. Um So um who thank you for tuning in to uh enumerate And I hope you have a good day. Bye.

Right. So, we're given this system and will determine its existence and uniqueness of a solution website should say of a solution. Mhm. But existence and uniqueness of a solution to the system. So, let's see, let's go and put this into a matrix. An augmented matrix form. He's got zero three. The X one is missing. It's zero. There is zero X one plus. Right? And then we got -6 And then we have plus six or positive six and 4 And augmented matrix is gonna have a negative five. And then we've got a three. Just make this bigger, negative seven and eight and -5 and eight. And then nine we got another three and we got a negative nine and we got a 12 and a negative 96 15, yep. That'll do series are augmented matrix. So, let's do what we can to get this into a row reduced form. And I'm gonna swap some rows here. I'm gonna get this zero this go down the bottom and swap it with the body with this bottom and bring it to the top. So, we got three nine 12, 996 augment 15 378. Next they eat nine. And then got 03 -6645. The matrix. All right. So, we're gonna clear out this robe the second row. So, we want to let's see, I want to add negative one times the first row to the second round. You might see it in some texas. The elementary row operation, you're adding like this negative one times the first trip to the second roof. You might see it like that. So let's see the first row is gonna stay the same. Somebody has the advantage of technology and just copy this paste much nicer. Look at that. So this comes out zero. So is he? It's going to be negative seven plus and negative negative nine. So it's evening seven plus that I'm not going to give us to And then we're gonna do 8 -12 is gonna give us four -5 -A -9. It's gonna give us four eight minus six. Can you give us a two nine minus 15? Negative six. All right. So now we need to cancel this. Go to this bottom row here. Mm Excuse me. The we have two leading columns in this leading entries in the second column which we gotta get rid of. So we got do you get rid of that somehow? So how I see this as I am going to go negative I'm gonna somehow make this to into a three. So take an elementary cooperation that takes Let's see it needs to be a -3. So -3. It was Because that will make that to turn into a three year multiply by -3 house. And I'm gonna do the same kind of thing copy this just for the sake of using the advantages of technology to do that. And this bottom row is going to be deleted. So I'm gonna go ahead and well it's gonna have zero there. Mm So this becomes -3. So then all right, there we go, that's zero. Great right in here with fractions. I'm going to get a little messy here. So this is times three have. So let's see. That's gonna give us -4 positive four. Excuse me positive 12/2. Oh look at that, it's gonna give us six. That's pretty cool. And then that's going to make that into a zero and then we've got this *** three have times this four. So that's going to give us oh my goodness, that's going to give us negative 12 over to negative six. That's going to give us uh negative six. So we add to the six. So that's going to give us a zero down here as well. Things are looking interesting people. And then we've got Our 3 -3/2 times to here. So that's gonna give us -3 that we had to four. So that's going to give us a one. We can't forget about this one, It's -3/2 times six. It's gonna be positive 18/2. So that's positive nine. He's gonna give us a four cereals. Can we clean this up? So here's our augmented matrix. That's been reduced And this so we're looking for the existence and uniqueness of the solution. So there is a solution, There is a solution because ah there are as many leading entries. The rank of this thing is three and there are three rows. So here we go. So there's a solution, however, it is not unique. There are actually an infinite number of solutions solution. Yes, but is it infinite? Or is it is it unique? Mhm. No, not you make there are an infinite number of solutions, so there you go.

Hello. We have problems by 11. Which is ready to the linear combination. So we have and the value of A. One A. Two and A. Three. And we need to determine if B. Is a linear combination of A. One. A two, A. Three. So be all even A. Two A. Three and B. Is are the mattresses so full these two billionaire combination. Yeah of B. B. Has to be equal to A. Three plus lambda A two plus meal. Even that is if you're if you can write B equal to in the linear combination of these three. So it will be in a combination of onate way three. Let's plug in. The value of B. Will be having to minus one six 835 minus 68 plus lambda 82 is 012 plus mu even is one minus +20 Okay so this is two minus 16 and if they just add it up it will become five plus zero into lambda zero lambda. It will become zero. Eventually plus um u minus six plus lambda minus two mule and eight plus to lambda class zero mil. Okay so from this equation if you equate these two it will become five plus musical to to which means Musical Too minus three. Miracle to minus three. So if musical to minus three, let us just get here to minus 16 equal to five minus three. That is to Okay minus six plus linda minus six plus linda. Just plug in. Just plug in mu equal to minus three. So it will become six. Yeah third eight plus two linda eight plus two lambda. So this will get cancelled out. So eventually it will become two lambda in eighth place to linda. So now linda will be called to minus one. Okay so we can say that B is written as a three place in the form of a. Three plus lambda A two plus you a one as be equal to a three minus because laminate minus one so minus 182 minus three because me was minus three. Mhm. Even so we can say that, yes, B is a linear combination of a one, A two and E. Three. Thank you.