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3. (15 pts) Let A = [2 ~3l8-[; prove vou dld these by handl#c-la #-lz SHOW SOME WORK,AV If Ais 3 matrix for lincar transformation T(x) , mapping R"+R", fi...

Question

3. (15 pts) Let A = [2 ~3l8-[; prove vou dld these by handl#c-la #-lz SHOW SOME WORK,AV If Ais 3 matrix for lincar transformation T(x) , mapping R"+R", find n and m.B) Compute, if possible: 2A ~ 38C) Compute; If possible: ABD) Compute, possible: CAE) Find P(D) if plx) 2x' _ X -F) Show the effect on the basic box in RR? of matrix D. Show what happens t0 the letter "R" Inside this box:

3. (15 pts) Let A = [2 ~3l8-[; prove vou dld these by handl #c-la #-lz SHOW SOME WORK, AV If Ais 3 matrix for lincar transformation T(x) , mapping R"+R", find n and m. B) Compute, if possible: 2A ~ 38 C) Compute; If possible: AB D) Compute, possible: CA E) Find P(D) if plx) 2x' _ X - F) Show the effect on the basic box in RR? of matrix D. Show what happens t0 the letter "R" Inside this box:



Answers

Exercises $22-26$ provide a glimpse of some widely used matrix factorizations, some of which are discussed later in the text.
(Reduced LU Factorization) With $A$ as in the Practice Problem, find a $5 \times 3$ matrix $B$ and a $3 \times 4$ matrix $C$ such that $A=B C .$ Generalize this idea to the case where $A$ is $m \times n$ , $A=L U,$ and $U$ has only three nonzero rows.

Problem. The worst 16 ends d one he to well before who are long story. There are. So, you know, for a is one X, it's were X Q. And it's for four and see in the sub several one So t two t one, uh, one. It is equipped to Why t two you want born what you use, he two one dash and that they don't want it here. So it's people, you know, site sequence who were or be to you one. Well, rats is equal to run. Teacher do you want for Rex? So it's equal to into It's the lash sequence. Food he to, ah, one currency war Teoh, Do you want all? It's where it's different You want? All right. Where in tow. Oh, excess spread. There's ableto Tito. Two deaths and excessive too. So different science smoking de to you want All right? You We went through Do you want or it's human. She different. You are. It's you. Yes, you to be X squared. They will be Times Square. She's well, partly toe Do you want of its or or just a quick will people or anyone or rex more for Jews Dean too. It's a border for Dash. They were to for its 434 times toe. Both these with people. Thank you. Really sure. These transformation and by an ancient organic or by nation Perfect as he. So, Tito, you wanna for in the Quito? Zero times one. And, uh, do you need to? If you want or X we put for one thing. Zorn you don't He won off. It's where it's April 4 times Run in tow. Anyone or excuse? It is different when times one and I need to You want our export for people to 32 times times one so t one people seeing it, Manny Different Well, one or room. And you coach in being you having fear 6.57 de to You weren't I don't see a I mean Steve be Do you want be a So he won from its inside 13 and 15. We get it. You won't be a different for four years, Warren. These years, the little two years, you know you know the ends you do you feel you know four And from here we know that t two c p is one, 24 and the these are from its was 15. 15. So therefore t one at 21 is the notification off to, uh, metrics Uses is equal to the one for well and 32. Now the question. See, you're going this year. Explain for five. Did to You want for me X? Yes. He wants to you toe you one for seeing replying. Valerie, be over Rex. Oh, great. And you've actually is to buy negative ones you like to be and don't you? Once you're breaking, this is what we get here. Be able 14 52 two of the most of it. Nation for things to matri, ces is is equal to 97. So computing directly real and that de to you want uh so that's fine. Minus X crew for three X card for is equal to two. Do you want to plus five x minus X squared three It's or or which is a book to you too. Are five minus two acts last run que is able to fine times Too much running by two us. Well, with the blind by Well, you put too. I mean several the same issue

For this example, we're going to be illustrating the the're, um that's provided here with our particular matrix A. And for this matrix, we know that Lambda 12 is two. Plus, I is an Eigen value, and the corresponding Eigen vector is V one with this, the're, um and the Given matrix say our goal in this problem is to determine the Matrix c indicated here as well as the Matrix P that gives this factory ization for a Let's start off with the Matrix p. To do that, we're going to focus on our Aiken. Vector at right is as follows. We can say that V one is equal to the rial part of E plus the imaginary part of E. And that would be first the real part. Let's put a night I hear is well, for the real part. We focus on these values that are not coefficients to the complex. You know, I So the real part is negative one and one that we add to that the imaginary parts of the complex number. For the imaginary part, we pull the coefficients of the complex unit I, which is a one here and from here we could say that the zero there is a zero times I to give us a 10 Then don't forget to copy the complex unit I. So this is what V one is equal to. And then fire the're, um here. We see that p is going to be first, the real part of E which is this vector? Negative 11 then the imaginary part of E. But that's the specter here at 10 So we found that Matrix P and what we can do to find the Matrix C is used this theory and provided here so or that portion of here, Um, we take C, and we'll also go to the Eigen value That's provided Lambda One is of the form of a plus B I. But in other words, we have a is to and B is one, and those values go inside here, so we'll have a 21 than a negative one and a two. So this is the Matrix P and the matric See that allows this matrix factories, ation, toe work

Okay, so phone from previous exercise, we have, um w is defined it as I minus seat hams. A minor's is I inverse times b, and we have fabulous times you because toe y and then we want to show. And so let's record at your own Gino system off equations. So this is the original system of equation. And the way observed that nablus is the sure compliments. Thank you to, um, a minus a science. So this is, uh, from previous exercise. But if we use another, um, substitution method So first we ride out the system of equations. So this is a system of equations we have. So you figure it's another reduction? Ah, answer it. Relabel this by vision one. The question to so fair question to we use another matrix, be acting on X, Knew it. We're have something like this, and this equation allows us to, um, glass us to cancel this you so we we can use the equation one minus three. So we have a miner's PC, minus his times. I n acting u X equals two minus B y. So, um, that gives us X equals two minus a minus P C minus a C I and immerse you Why So with this result equation for we can park him back to equation, too. That gives us minus see Time's a minus. B C minus iss I and diverse B comes Why plus you equals toe. And, uh, we just moved the first term to the right. With directly off 10 solution for you, you equals toe I plus C terms a minus p c minus days. I m immersed times be why? So this result shows us this term it's actually euros off w ways because we know that w is you you caused to Why, that means you he caused to tableau is in worse off. Why so Diarrhea? Zimmer's has to be this term. And, uh, this term is the shoal is the sure compliment off the following matrix which finished the proof in this for this question.

Hi there in this occasion. We're going to work in the vector space of poor animals of utmost degree one. That means the space P one. We're going to consider to vacuum to basis for this space. The basis be that's going to be generated by the polynomial P. One and P two that you can find here. And the basis be prime. That is going to be composed by the point of Mills Q one and Q two that are defined below. We need to find first. The transition matrix from the basis be primed to be. What is the procedure? Will we need to consider next and extended matrix or first? We're going to put the new basis in this case the bases beat. And in the extended part we're going to put what corresponds to the old vases in this case the prime. Okay. The idea here is that This we need to write this point on those in the standard basis for P one. That means that in this case P one that is equal to six plus three X. Well corresponds to the picture 6, 3 In the standard basis for P one. Similarly, P two will be the victor. That is the .9 of 10 plus two X. Is the picture 10 to The victim. The point on your Q one in the more space here. Q one. Well, that is equal to two Is the vector just 2. 0. And cute too. That is a polynomial three plus two. X Will correspond to the Vector 3. 2. Okay. And we're going to use this back to representations in the basis. This coordinate factors to construct this extended matrix, that I define it here. So the new basis will corresponds to be. So we need to put here 6, 3 and 10 to in the extended part we need to put the vectors to zero on 32. Okay, so we have now or extended matrix. And what we need to do is reduce this matrix to the asian form in such a way that you obtain in the left part, the identity matrix. And in the extended part you will obtain the transition matrix from the old basis to the new basis in this case, from the prime to be. So that's what we're going to do. We're going to reduce this this matrix to the action form and to do that first we need to put a number one in the 11 entry of this matrix. That Can be achieved easily but multiplying the first row by 1/6. So one five thirds, one third on one half 3- zero and 2. Now we need to put a zero in the entry 1, 2 of these matrix. That means that we need to assign to the 2nd row. The second row minus 23 times the first row. So the first row will remain the same. And here we're going to have a zero, Then here -3 -1 and one Health. The next step will be put one in this position. That can be done easily. But my multiplying the second row by -1 3rd. So our matrix will become 110 here, five thirds, one third, one health one third And -16. Finally, we need to put zero in this position of the matrix and that can be done by multiplying the first row by taking the first road and obstruct him 5/3 times this second row. And now we have here on the left part, the identity matrix, that is what we wanted and in the right part we will obtain the transition matrix. That in this case is it was to to over nine, 7/9, 1/3 and -1 6th. So this is our transition matrix. We're going to regret this as from B prime to be. Yes. The transition matrix from Victory prime we're going to take Is a common factor. Why not? 19. So that we obtain here More writable matrix that these -2 seven three and I'm in here. Well, let me check yes minus three health's Or well, I think that this better 1/18. And in that way we will obtain the matrix here minus four, 14, 6 and -3. They think that this looks better. Okay, so this is our transition matrix from the basis. Be primed to be for the next part of this exercise we need to calculate now the transition matrix from B to B prime, but that can be obtained easily by taking the members of the previous matrix. That means the members from the transition matrix from from B prime to be will be this transition matrix from the basis B two B prime. So we just need to take the members of this matrix 1/18 times minus 46, 14 minus three. The members of this matrix, what's going to be that members will? The members is equal two, one over four times three, 14 six on four. And this will be the transition matrix from B to B prime. Then For the next part we're going to consider the following polynomial. The polynomial is equal to -4 plus x. We need to grate this in a coordinate factor in the basis beat. So what's going to be? Well, first, Well, to to to obtain this, we're going to write this first as a coordinate vector in the standard basis of the polynomial. That means the vector -4, 1. And we need to create this vector as a linear combination of the element basis in B. That means this is equals to all for one times the first element in the basis B. That is the vector 63 Plus α two times the vector 10 to when it to solve For Alpha one. Enough for two and we will obtain that of one is equal to one And that Alpha two is equals 2 -1. So the coordinate lecture of this point of melty in the basis B Will be the Victor 1 -1. And in order to write this this polynomial in the basis be prime. We need to multiply the previous vector by the maid transition matrix from B to b prime. Okay. Times deep coordinate lecture pete in the base is big. So these matrix was equal to 1 4th times three 14 64. And times in this case by the picture one minus one. That is the this polynomial in the basis be the coordinate vector of of the paranormal in the basement. Okay so we just need to multiply these pictures this picture by this matrix and we obtain that the polynomial the recording intellectual of the polynomial in the basis. Be prime as equals two. 1 4th times the vector minus 11 and two. And of course we need we can verify this directly. This is using the uh the transition matrix from the basis B two B prime. But we can do we can perform this directly. So let's regret. People was equal to minus four plus X. And the element basis in big prime. We're the we're the plan our meals too On 3-plus 2 x. Okay so again we can represent this as vectors to make the things easier we have here for one and we need to grate as a linear combination of the element of the element basis in the prime. That means you're all for one Times A Factory to zero. Let's put it be better. Better one plus B to to. And the victor here. 3, 2. And the solution here for B to one is minus 11/4 and forbidden to east one half. That implies that the the coordinate vector for P. In the basis be prime is the vector -11/4 And one half. Or what we obtain here, 1 4th Times The Vector -11. And here, too, just what we obtained in the previous part.


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