in-mathbbp_2-find-the-change-of-coordinates-matrix-from-the-basis-mathcalbleft1-3-t2-2t-5-t2-12-trig

Name: in mathbbp_2 find the change of coordinates matrix from the basis mathcalbleft1 3 t2 2t 5 t2 12 trig
Uploaded: 2019-08-10
Duration: 5 min 36 s
Channel: Numerade
Description: In $\mathbb{P}_{2},$ find the change-of-coordinates matrix from the basis $\mathcal{B}=\left\{1-3 t^{2}, 2+t-5 t^{2}, 1+2 t\right\}$ to the standard basis. Then write $t^{2}$ as a linear combination of the polynomials in $\mathcal{B}$ .

Question

In $\mathbb{P}_{2},$ find the change-of-coordinates matrix from the basis $\mathcal{B}=\left\{1-3 t^{2}, 2+t-5 t^{2}, 1+2 t\right\}$ to the standard basis. Then write $t^{2}$ as a linear combination of the polynomials in $\mathcal{B}$ .

Runpeng Li · Accepted Answer

first we have the Colleen Ami o Space and we first have the basis that is B, which is one minus three t squared, two plus T minus five. He&#x27;s square and one plus two tea. Okay, so the standard days, that&#x27;s just a key one. This one you do is two. Three. Is he squared? So if we transform those constant and call your patient and two squared by by a P want Peter MP three So we have basis be is you want minus three the three. Two p one us. You do minus why p three And he won t o p. Two. So that gives, uh, change. Accorded a matrix from E to see to see to be first. 10 negative. Three and 2165 and 120 It&#x27;s so here&#x27;s dollar matric. So the next thing we need to write t squared as a wiener combination, not the point. Normalcy B. So that is too. Find a p from E to see times record be e and record UFC and work. Even pfc to is the victor 001 and we will solve this system. 121 There were 12 and the three negative. Five and zero and 001 Here&#x27;s the system we need to solve. So we apply the gushing in the nation two. Why don&#x27;t we? I&#x27;ll just write the result here. That is 100 They&#x27;re all one deal 001 and have three negative to one. So that gives, um, Specter record. PB is three negative too. And what? So our we need a combination is that is three times the first in your question one minus three T squared, minus two times a second question two plus T minus five. T squared to us. T minus five teeth were or we had we say t squared. Where you either We can see three and plus one times off one of us to tea, and we can&#x27;t even check. We have, uh, three minus two. Oh, wait. Um, yeah, we have three. Why is two times two is negative for and plus one. So that&#x27;s countenance with zero. And we have that give 60 squared and and positive 10 t squared and has tools when that years. And yeah, we have, uh So then will leave us a T squared and we have to t cancel two by two t like duty here. So that is exactly two D. Oh, Sorry. Uh, second U T squared.

Runpeng Li · Answer

Okay, so for problem for I r. Again given assumption we have t of x one, be one of those x two you two plus x three The three It&#x27;s the new vector, which is to x one minus four x two was five like three and ActiveX too. Plus three x three. So to find out the matrix for tea rented to be first need to find out the f B one from court in a C where C is Thea faces off off are too. So here we can just take X one to be one and next to x three to be zero. So our specter will be too zero. And similarly, we can find bee the F B two. So this will be negative for Nick to one and finally t f B three so T o b three Just be five and three now. So the Matrix TB, we&#x27;ll just be the span off these three vectors. Two negative four and five zero negative one and three

Doruk Isik · Answer

Yeah, in this trouble were given a set up. So we&#x27;ll start solution by writing the organ directors for each polynomial. So for people in the quarter, Inspector Ho. But like this 101 said, the 1st 1 is the question. But these year old 2nd 1 is to one of you two. So using this Pete, too apartment that too perfect. You could be rich. Zero. What being? And for P three, that could be, um, that would be one one negative three. So then the augmented matrix phoned my bees. Corn factors would be 101 01 negative three and 11 Negative three. Now let&#x27;s write this over mental midgets in traditional echo form. And that is he could do warn. 00010001 is you can see. We have three columns and each column has a few bits elements. So three columns, three events elements at each column. So it means that these vectors Spector&#x27;s spend our fee. It means that the given Colonials no meals spin he two. Now in part, we were given 1/4 collector for a loan Jew, and that is next to 112 And whereas to find this bologna to using this So we know that cube of native one times people plus one p two plus thio three So Dunn will be negative one times ones You&#x27;re one plus one times zero wounding three waas Two times One wonder three And we find this to be one the next 10. And using this, uh, using the information that these numbers represent corruption tease. We can ride the roller milk you is one plus B t minus 10 t spring.

Arsalan Afkhami · Answer

we need to find the change of basis metrics from a basis be to basis. See? So first I write the first full union off order basis be and then need to find a victor. Um, the three components, C c one c two c three. That many. Smith replied, Why the criminals off or there? Basis sees going to give us, um, this, uh well, you know, so see, one guy by first event takes us see, too Supply by Yeah. Native Express, X two plus C three one plus two. Experience too. You to rewrite it based on excess. Uh, C one plus C one. Thanks. Plus seats here. So thank you. So maybe see two x plus C two expert two plus C three plus two c three x two. So fresh the same problems Here We have an X here, then exposure zero. So I rewrite it based on the powers ex to move bye bye. Minus C se us first use here. See to plus two c three this extra supply by sea one minus C two us. What&#x27;s left? So C one plus c three year hair, Same unita. Great. The same powers the coefficients. Here we have one. And then here he&#x27;s There&#x27;s no exercise your most by by ex year, so it&#x27;s zero. Here. Use one on here, these two. So to on. To find C one c two c three, you need to find it&#x27;s just stuff equations. I write c two plus two C three because 21 see one go minus C to physical stores ERA and C one plus C three musicals to to it&#x27;s easy to solve. Here we have C one because to see to then the you play seeds thes two equations So they have, for example, here you can write See you one plus to see three close to one C one plus C three to now we have this system for operations we to, uh, equations. And for some baby right? See want these because two to minus C three and then we solve a wrist. So now we can easily find C one c two c three, which will give us the first victor. I would like to see just going to be 33 and media one. No need to do the same for the second, pulling you all right? The 2nd 1 If he&#x27;s one minus negative one on the six x plus eight Explorer to is a cause to She won as before one plus x plus c two x six to plus C three, one plus two x two. So, as as we did before, it&#x27;s going to be the same as our above. So if I&#x27;m not reiterated again, I&#x27;m just going to write this system of equations here, as we have here. So just rewrite it here. See to us to see three c one c two on C one. She won C three on the here. You need to check the coefficients here, here, here. So there are other things that changes you. Have you have eight six negative one. Then sooner before we saw it, I find see Wannsee to see three. Then we will have his second rector that you&#x27;re looking for to see and to the before 23 and then we need to write this said, Well, you know, which is seven three picks minutes nine x two Because to as before, see gone through a one plus x must see to try by I mean this x plus x two I see. 31 plus to explore virtual. Ah, same C two plus two C three, as I wrote above. So you want to see three mill here? These are called to 93 son. So All right, 93 Soon on. Then we need to solve these. Fine. C one c two c three, maybe solve it. You have third. Victor has nice to one. Made its size. So here. These are the columns off the metrics we&#x27;re looking for. So all right, be change of basis Metrics from B to C C. Cold. The columns. She&#x27;s 33 81 before to three. You too. 1 85 This is the metrics were before

Runpeng Li · Answer

okay for problem Problem 10. We have the leaner, faster mission maps from P three to Arthur are poor. Such that t of pulling nominal is equal to by taking different input to the Polo Meo. Just pee of connective three first p A connected one and p of one and p up three. Hey, So the first thing we need to show that is that, um to show this transformation is linear. Okay, so do that. My transformation. We can first check tee up. People ask you where panic you are all clean on yourself. Order to be so then, by our assumption, we can have a p rescue while we take the include to be elected three. And he ask you with the input elective born people ask you with include one and people ask you with your food three. Now, because we are adding these two point nominees so we can we can calculate the number Sweet that with this input on dad them together just just by by the rules, our tradition. So that&#x27;s he, uh, 93. I trust you collected three and PR. Negative one. Uh, you, uh, negative one and p of one less queue of one. And here, three askew on three. So by now we can separate this, uh, this vector with by a vector off Minami O P and the vector polynomial Q. So it turns out to be thio he that&#x27;s t l Q. So that&#x27;s the first part here. So the second part is considered skater Linda times people. So again, by our assumption, we still have Lunda cons. P off negative three Lambda SPF connective one, huh? Times p of one and Linda Times p of three. Excuse me. So we can take out the skater from our rector&#x27;s. So we have loved a in front of our specter and P uh, negative three. He, uh, connected one he won and ps three. So that turns out to be Lunda time. So? So we&#x27;re done for the first part because I our this relation and second relation. Then we can conclude that he&#x27;s a renewed transformation. Now, Part B. We need to find the matrix for tea relative to the basis won t t square t t cubed for p three and standard basis are for Excuse me. So now let&#x27;s, um defined business P start this is B to be given basis in our assumption, which is one two he squared and t Q. So we first calculate, uh, actually the tea with the input up won t t square and teeth cute separately. So first we have to be one which gives our vector all once and then we&#x27;re plugging t which gives Dr Inactive three negative one, one and three. Then we&#x27;re plugging our t squared. This gives Excuse me. Um 911 night, and at last we plugging our thank you. So we had a vector collective 27 negative. 11 and 27. So our metrics will be this band of these four vectors. So this will be he of this is B should be one collective three nine Next 27 one defective. 111 1111 and 139 27. And that&#x27;s it.

Ernest Castorena · Answer

for this question, we have another subspace, uh, are three in particular. It&#x27;s the set of all vectors defined by some questions, and they want us to find a basis for that subspace. Well, it&#x27;s all vectors. That is the solutions to these equations. Uh, let&#x27;s first map it. Teoh the tools that we&#x27;re familiar with, like a system of equation him. That&#x27;s right, Our equations. Pero, Well, it&#x27;s our three. So we want undue three by three warm. And I feel in for this. We&#x27;re already here just so we could easily see what the subspace should be in terms of the domain Won&#x27;t this tells us the intervening back into equations we have X is equal to the wine track. Seven We have that y is equal to y Z is equal to see good No im gration yet well, the rank of this matrix is just one, so it&#x27;s no space dimensions to, But the no space is also the solution to the system of equations, so it&#x27;s banned by two vectors. In particular, we have to free variables. The why on the Z and X is dependent on them from how he wrote it, for how we sold our system X has to be negative three of whatever wise Andi Z has to be our excess to be one of whatever z isas well, and you&#x27;re able to add these together to get another solution, be able to scale them and getting initialization and Adam so thes two vectors form a basis for our subspace that

Problem

Given $\mathbf{v}$ in $V,$ there exist scalars $x…

Problem 14 Hard Difficulty

In $\mathbb{P}_{2},$ find the change-of-coordinates matrix from the basis $\mathcal{B}=\left\{1-3 t^{2}, 2+t-5 t^{2}, 1+2 t\right\}$ to the standard basis. Then write $t^{2}$ as a linear combination of the polynomials in $\mathcal{B}$ .

Answer

$t^{2}=3\left(1-3 t^{2}\right)-2\left(2+t-5 t^{2}\right)+(1+2 t)$

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$t^{2}=3\left(1-3 t^{2}\right)-2\left(2+t-5 t^{2}\right)+(1+2 t)$

Linear Algebra and Its Applications

Lily A.

Catherine R.

Heather Z.

Caleb E.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Lily A.

Catherine R.

Heather Z.

Caleb E.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications