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Consider Pi, the Lincar space cf polyuctnials f (t) of degree nt Dsst 2. [bri] What is the dimeusion of P;?(6) [Spt] Show that the map T P+ Pz given by TU(C)) = /()...

Question

Consider Pi, the Lincar space cf polyuctnials f (t) of degree nt Dsst 2. [bri] What is the dimeusion of P;?(6) [Spt] Show that the map T P+ Pz given by TU(C)) = /()- f(-0},ia Zicizn: Crcruionnitko[Spe] Pird the watrix of T , uer the rais {L1+61+t+P} o R:

Consider Pi, the Lincar space cf polyuctnials f (t) of degree nt Dsst 2. [bri] What is the dimeusion of P;? (6) [Spt] Show that the map T P+ Pz given by TU(C)) = /()- f(-0},ia Zicizn: Crcruionnitko [Spe] Pird the watrix of T , uer the rais {L1+61+t+P} o R:



Answers

The transformation T maps [x // y] to [a & b // c & d][x // y]
Show that invariant points other than the origin exist if ad - bc = a + d - 1

In this question, we are asked to draw. Ah has a dragon for policy It off? I said s with four element with put your order as being a subset. So for element the power said we have 16 in women, right? Two to the fort and we got to draw them all. But I already finish it. But I will explain it a bit. So it looks like this Sorry for a messy lie in the middle. So we have four lay years. Sorry. Five layers. And you You can see that it has a pattern. All the number off elements, right? This flow has zero. This has won 23 and four elements and ish issue lie rivers in being us offset to the to the next floor. So and they said, is a subset off. Oh, every set here. So it has. These numbers are the number of like, coming off each subset, right? This has four line. This floor has three lying issue. So going out like this, this lie has to lie ish, and this has won. And there is not much to say other than just don't be confused by like it may have a lot off subset. But the principle is just that when when something is us upset all and that does it, then you just draw the line. That's all there is to it. You have to listen all them in this fashion, because for each floor you go up a step on each floor. You have relations such that there is nothing in between them. So there is no like subset that the can map to and then map to CD again. So there's nothing like that. This way to the has said again. That is it for these questions. Thank you.

Problem. 30 Question A jack O. G. Is equal to the determined off D X portion. Ex portion partial x portion you which is do you over me. Partial X Portia lvy, which is negative. New square over the square, partial y partial you, which is negative. B squared over you square for sure. World portion ability is toe be over over you. So the reason for this determined is this item Mind that this component this is not the blind by this component minus justification for these two components. So the final answer for that will be four minus one, which is three, which should be the image, uh, one toe four. Enter the Magi is G one, the fall, which is one over cool and 16 over once, or is equal to 1/4 and 16. So the threaded part is is deep where this is Why access and it is the x x es. And this is the shredded port, which is three where this is 1/4 and this is 16 and this is 1/4. And this is 16

So we have a precision that is even by deploying expression seeks signed to t along the either action You are six goes well to be from the J Direction two d Jianye direction Plus why d no que aviation about the compute wasti your normal The one discover curvature on what he's, uh the unit normal. So let's compute first. What is V So in friendship these So there is little sign signs of because the the baby travel Get out to them 60 well crusade thing I from here you have a minus sign Good tea Now then by the channel your inner six times 2 12 one day and there's five d just five over the cave. Original bodies over the J direction. So here again but cattle there so co sign off t I. I signed 30 j lost five so that it's no people do the square root off these squared Sapele Square will sign I want to d squared Was the principal components squared plus well squared Sign off to t squared Was that wanted squared Those five squared Teoh here we conducted a poll I will sign up to d squared 12 times 12 square damn school side of the square because I'm a duty squared Waas sang off 30 squared. That's fine. So something verification here is one. So be well the square. It's fired square. Um, so this fire was quarries. Do you live and filled the square one for you? More. So this would be Discworld or, um 1 44 I lost 25 nine of being for so see one six mile, which seems to be 40 three because so all these number varies. Oh, squared. Uh, where we, uh So these, uh, Morgan live with us squirt of warm 16th of disease. Eagle do 16. Ah, that's them. Waas three squared from the You can just be like that. So square the 1 69 So, uh, so that t will be equal to, um, these vector divided by disclosed 1 69 So Well, who Sino two. D. All right, I my no sign. Oh, to t j not divided by a squared. Oh, 1 69 Plus, um, my okay. R squared with one 69. So this is he I'm that is gonna be done. Is gonna be e e t. The very one. We're gonna build a lost huge. There are relative disease, so we have to compute What is empty? Yeah. So, uh, well, the key, uh, will be the compactor. 12 for screwed of 1 69 So it's great. The one sees nine from being really harsh. These so? Ah, two times minus sign to D I That being two guys co sign my bicycle sign school dams minds were saying to here j and then for day these we have really at their resume. So just be just that when we come factor do there she would be 24 doctoring You do? So I'm still bye bye squared of 169 one better. We have a minus. Minus sign two d I plus co sign T. J um so bodies the TT And then what is It's norm also very would have squared of diet both this girl So 24 squared 1 69 The times sign duty squared plus rules to do square. So all of this part will be one. So it would be for 24. He's embraced more over screen one 69. Well, so that, uh well these couple will be 24 divided by a squared with 1 69 onda. Uh, the exclude them a normal fees Screws of 1 69 So to that Well, like these to the four over one 16 night so that he's, uh that is a curvature Kate on the no. The normal for people to t u t divided by its Not so. Is this these vector so Well, huh? Minus 10 4 over square you go. 169 Uh, yeah. Times sign of I Sino to t I Law school Sign off to t J from the Nazi divide. But ah, the norm is this. But why did he or were? I want 69 so that we see that these canceled giving us one on one So it is normal should be equal. Teoh Mine Sign of pity. All right, Miners co sign off to t Jay. Really? The well you can check Berna some simplifications. Well, so So this deal. Oh, yes. So as you can check here Well, is this components plus that from planet? But you can check that you didn't the product off t you then Hewitt an The two components her are flu was it happens. Ah is the product should be should be zero, which it is so

The transformation T maps The Vector X. Y. two. The product of the matrix times victor. The metrics is A. B C. D. The vector X. Y. That is the metrics ap city times to victor X. Y. Is the transformation of the picture X. Y. Through the map. T. Or its information. T. Show that in variant points other than the urgent exists if eight times the minus B times C equal A plus the minus one. So these information T goes from victor's of two components are too From Victor's of two components because the Metrics A, B, C&D. These two times or two x 2. And that multiplied by two times one victor give us a two times one picture. Anti of victor, X. Y. Is to find a symmetric ap C and D times a vector ex wife. That is. We multiply the metrics here times to give him victor and that result that result in victor is the image of the victor X. Y. Given. Now we talk about in variant points also called fixed points. These are points that has the property of having as the image through its information T equal to the same point. That is very important X. Y. In the plane as Birdie. That T of the point is it will the same point. There is the point is not modified when we apply the transformation T. It's called variant point. It is clear that the origin is an environment point. Because if we multiply the matrix times 00 victory. That is the origin. We get again to Syria zero victory. So 00. The origin is an in variant vector or any variant point. So we want to write the condition that need to be needs to be fulfilled in order to have in variant points other than these. Seriously a picture. So let's say that T of X. Y is equal to X. Y. If and only if T F X Y A B C D times X. Y. That's the same. Sorry, equal to X. Y. That is because of X. Y is defined this way. This is equivalent to let's multiply the metric stein's victor. We get a X plus B. Y. His first component, second component C. X plus dy. And that can be equal to X. Y. And equality of vectors means that correspondent components can be equal. So we got to have two equalities, E X plus B. Y equal X. And see eggs plus D. Y equals Y. And put in these terms to the left of the equations. We get a -1 Time six. That's when we put eggs to the left and take a factor eggs plus B. Y equals zero. And here with the same with Y we get C X Plus The -1. Y. people soup. Mhm. Okay. And this is the same as a matrix vector system. That is a linear system in X&Y. Which can be written a metrics form as a -1. Me first role of the metrics than C. The minus one times X Y equal serious. Ooh so this is a metric form of this linear system here. That is we want to find X. Y. Different from the version which satisfies his equations. So to have a non null solution to this equation. This linear system, it is uh equivalent to saying that the metrics is not invariable because if the metrics a no swan pc and even this one is variable multiplied by its inverse to the left and the only solution will be xy consider so. Okay, let's say that this way here linear system Amen, swan, E C&D -1 times X. Y equals serious. You as solutions different from 0000. Sorry if and only if The Metrics A Menace one, B. The U -1 is not either do has not been verse and that's the same. And this is equivalent. Two. The determinant of the matrix equals zero. Okay, equivalent to the fact that t determinant of these metrics be equal 20 And the determinant is not zero. The metrics is invariable. And as we said here, above the system, we have only the serious, serious solution. So we need this determinant to be equal to zero. That is the case in that case mattress has no reverse And this system get to have solutions infra France 00. There is determinant of the matrix. Amen is one. B C T -1 equals zero. Mhm. And we know this determinant is equal to the product of the elements on the main diagonal, A -1 times t minus one minus the product of the elements on the second, f minus B C. And that gotta be equal zero. And so we do this multiplication, hearing at a d minus a minus D plus one minus bc. He got zero. And so a d minus bc. That is left is to terms on the left of the equation equal. And the three terms go to the right of the equation. We get a plus the -1 which is just the condition said uh Name in the statement, there is this one here. So if this relationship between the interests of the metrics, A B c D defining the linear transformation is satisfied, then there will be in variant points of this transformation or to this transformation different from the origin. And that is because uh when we stayed the fact that it victor, x wire point X, Y. It's an environment point which is this equation here. This lead us to a linear system which will have non serious solutions. When the determinant of the matrix is equal to zero. That is when the metrics is invariable, it's not over. So the determinant is calculated and leads to this condition between abc and the Okay


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