the-set-mathcalbleft1t2-tt2-12-tt2right-is-a-basis-for-mathbbp_2-find-the-coordinate-vector-of-mathb

Name: the set mathcalbleft1t2 tt2 12 tt2right is a basis for mathbbp_2 find the coordinate vector of mathb
Uploaded: 2020-12-10
Duration: 4 min 19 s
Channel: Numerade
Description: The set $\mathcal{B}=\left\{1+t^{2}, t+t^{2}, 1+2 t+t^{2}\right\}$ is a basis for $\mathbb{P}_{2}$ Find the coordinate vector of $\mathbf{p}(t)=1+4 t+7 t^{2}$ relative to $\mathcal{B} .$

Question

The set $\mathcal{B}=\left\{1+t^{2}, t+t^{2}, 1+2 t+t^{2}\right\}$ is a basis for $\mathbb{P}_{2}$ Find the coordinate vector of $\mathbf{p}(t)=1+4 t+7 t^{2}$ relative to $\mathcal{B} .$

Priyanka Sadarangani · Accepted Answer

okay. In discussion. Settle cases B is equal is given one plus t square, deep, lusty square and one plus two t plus thes. We&#x27;re okay and is the basis off p two and find the coordinate vector off. Beauty equals to one plus 40 plus 70 square related basis. Okay, now can. Now we can write this in metrics. Thomas de Squire tea and one so we can write this. Let one from this day, Squire. First, we have to write this T square one in the coefficient off zero and constant one. Now, from this P square is one. He is one and question zero from this D Square one, the coefficient off piece, too. And constant one. And the solution is seven, four and one. Okay, step by. Step on. We have us all this now, even swallow this. Okay? So fast. We have to make this part zero. So we will do. Are three r three equals toe one minus are Okay, So what we get is 111 sailing 0124 on 0106 Okay. And now now what we do on one needle? No, again. They will do this portion zero. Okay. And for this, we will do R two equals toe are two minus artery. So we get 111 70 0 to 4 minus secrets minus two on this will be same. 0106 Okay, now what we do, you do simply now we will find out, uh, are to divide by go only Why get 111 seven 00 You were too. Is one minus one on a 010 six. Okay. No, What we do is r one equals toe are one minus R minus artery. For that, they will get When Zito Zito 2001 minus 10106 Okay, so now we 13. Our goal with our three. So what? We get 1002 0106 on 001 minus one. Now we have the standard metrics. Okay? And what is our answer? This is over. Answer. The answer is toe six minus one. P. B equals two. This is our final answer. Thank you.

Doruk Isik · Answer

in this problem, we&#x27;re given a set to be, which is a, uh, basis. And we know the full meals that form this base. So we have people are p two mp three in this problem where has to find the coordinate vector that we need to get this balloon GOP off too. So let&#x27;s assume not be coordinating Director. The cell performs. He wants Regency three. So what we know is this you know that c one people plus C to pee too. Plus e t p p should give us given p below me. So we have C one times one minus two squared plus C two times t minus. He&#x27;s very waas. See three times to minus two t plus t spread is a good three plus t minus 60 spread. So we have three unknowns. He wants you to nc three, but we have three new creation, so we have C one less to see. Three busy bee c one plus C 21 c 36 and we have C two minus two City is one. So using this we find see one to be seven you don&#x27;t see too to be negative three and CP to be negative too. Said so it on the coordinates back to that we&#x27;re looking for is seven negative beat and negative too.

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.

Runpeng Li · Answer

work. Given the faces, that is, um, one one minus t. And to minus 40 us t squared. Yeah. So we have these three a nominal size. Our faces. So we honored to Right down. He up t in terms off. Coordinated back. Factor off P lifted two spaces. First down, Petey is seven miners on a safety plus re t squared. Get less. Dean owed one as p one and one minus TSP too. And do you mind us? 40 plus t squared as p three. So in order to get three t squared, the only the only way to get people can t squared is too is by using the p three. So at least we have to times p three by three. That way we can get three t squared minus well, t, that&#x27;s sick. Only in this way we can get pretty squared. Otherwise, we don&#x27;t have t squared in p to m p one. So And you too. We need to combine Pete even p three to get connective and activate e. And we What we have for Peter is negative t and we&#x27;ve already had we Robert He already had a negative 12 so the only thing we need to do is two times you to buy electric port so that he so that this this, uh, this is equal to negative for last 40 or teeth. Okay. So that our, uh, Osama off three p three milers, four p two will be well, give us negative 80. So, in terms off and then ask in terms off a constant term, we have one. But previous previous stabs our already give us, uh, to because six minus four would be too s o. The only thing we need to add is fight so that to minus to plus five will be seven, which is here. So our final answer will be first three p three minus 42 plus p one. Well, five, if you want. All right, So this is our by no answer.

Runpeng Li · Answer

pick. So, in this problem, we&#x27;re already given the, um, Helene Alina transform Matrix relative to be, which is a three by three matrix. That is the real inductive. 61 05 elective one one negative two and seven. Now the only thing we need to do to find out a t of three of the one minus for B two is to take the coefficient off the given victor three V one minus four Be too. So it&#x27;s three off the one negative for P 20 for B three and enter the cleaner transformation off this matrix and we find that is 24. Order for century and active 20 for a second entry and three waas eight. You love it for the second entry. So that is actually the tea. Uh, three three b one. Honest for Pete to

Millie Lopez · Answer

Hi. So, for this exercise, we need to find the Matrix for the transformation off T By the given basic vectors that artie t one t two we buy knowing that we can start by finding the images off each one of them after their leaner mapping anti. So we start by Do you want equals to t He&#x27;s gonna have a multiply by there basic vectors that it&#x27;s one t anti too because of we&#x27;re looking sorry. So we&#x27;re looking for t one. Our values are going to be centered on one. So one is going to be one t is going to be Cyril. NT two is going to be also Ciro Ang. These values, we can change them too. A Ciro A one and a two. Okay, so in our general equation, we&#x27;re gonna be So did I in each one of them for defining off the leaner mopping. So we start by tree a Cyril, we have the brother that he&#x27;s one. So we right here, One applause we have here 50 So five also gonna be multiply by one. But then we have minus two one minus two a one so minus two, we have the value off a one that is zero. So we ride that to its multiply vice zero later. We right here that they have tea. These right here. Then we put a clause for a one that we have for a one. A one has the value of zero. So for want to buy Vice zero waas 82 That also has a value of zero. So we right here zero close it and we right t two Now we go tree stays the same. So east tree Here we have five minus two buddies zero So here we ride five, but its multiple righty So is five t here we have C room and Syria also here. So this is going to be more to buy vice zero. So this is going to be zero to you in leaner mopping. This is right then, like tree five. This zero. That&#x27;s our first call him. So you go for our second that we&#x27;re now gonna use that t won t t Sorry. You write the equation that equals two t one t t two. So we now want to find a value off t. So now our one is going to be 80 Our T is going to have the value of Warren anti to also going to be zero. So now we do. The same thing is going to be our 80 a one and a two We started. We write the equation, so he&#x27;s going to be t t equals toe tree 80 waas five a Cyril minus to a one old is wanted by by t one applause four a one pose a to And this is more to buy by tedium. Okay, so we now simplify it by we have the value that treat one 380 is going to be zero. So we right here tree zero five a zero Also gonna have a multiplied by C room to a one these time is going to have the value of one. So we multiply by one and we write our tea. A one for a one also has a value of one. So he&#x27;s going to be four. Don&#x27;t supply by one applause A to also has a value of zero. So we write zero and we hear Multiply by t two. Now we go here has the value of zero. Here we have the value of zero. But here we have negative too. So is going to be negative to here we have four. And here we have zero. So we&#x27;re going to write it like zero minus to T plus for T two in a linear mapping. This is also right and lie zero negative too. And four, then for our next one is going to be t equals t two. So this is going to be able to t multiply by one t t two. This time we want to find t two. So one gonna have a zero and t is gonna have a zero, and teacher is gonna have a one. So we right here zero zero one. These are gonna be our values for they want a Cyril a one on eight to So let&#x27;s start with the equation again. Three tree a zero waas five a zero minus two a one all multiply by t applause for a one because a to and this is going to be multiply by t two. Okay, so now we served the type the values a Cyril&#x27;s gonna be here. Ciro a serum again Here. Zero a 10 a one savvy Cyril here is gonna be a to one. So we right here, three won&#x27;t by by zero five multiply by zero negative to want to buy also via zero, for it&#x27;s gonna be also multiple. I zero and we have here alone. Plus want It&#x27;s now the value that this is gonna give us this is gonna be a zero is gonna be here, Cyril. And here&#x27;s zero. So we right here, Ciro plus zero t we have here that is going to give a Cyril but is plus one. So one multiply by t two. It&#x27;s gonna be plus one berated like this so it could be different. T. Teoh, this is writing in Poland and Lander. Let mapping is gonna be zero Cyril and one Now we have all the values we have. This one. We have this one and we have that one. So now we&#x27;re gonna with them build our matrix. So now we write the basic vectors coordinates as the columns for our matrix told that May right here Matrix is gonna be am And then it will be then basic matrix for the linear transformation is going to be equal to do you want T t and t t two. So on t one we had the values off tree five and zero on t t. We had the value off zero negative too. And four on t two. We had the values off zero zero and one and is going to pay the final answer. Oh, my God. This is going to pay the final answer. Thank you.

Problem

The set $\mathcal{B}=\left\{1-t^{2}, t-t^{2}, 2-2…

Problem 13 Easy Difficulty

The set $\mathcal{B}=\left\{1+t^{2}, t+t^{2}, 1+2 t+t^{2}\right\}$ is a basis for $\mathbb{P}_{2}$ Find the coordinate vector of $\mathbf{p}(t)=1+4 t+7 t^{2}$ relative to $\mathcal{B} .$

Answer

$p_{\mathcal{B}}=\left[\begin{array}{c}{2} \\ {6} \\ {-1}\end{array}\right]$

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

Linear Algebra and Its Applications

Lily A.

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Caleb E.

Michael J.

Vectors Intro

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$p_{\mathcal{B}}=\left[\begin{array}{c}{2} \\ {6} \\ {-1}\end{array}\right]$

Linear Algebra and Its Applications

Lily A.

Catherine R.

Caleb E.

Michael J.

Vectors Intro

Vector Basics - Example 1

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

Lily A.

Catherine R.

Caleb E.

Michael J.

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

Linear Algebra and Its Applications