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Let B be the basis of Pa consisting of the Hermite polynomials2+412 and 12t + 8t3 _ and let P(t) = 1+ 412 _ 8t? . Find the coordinate vector of _ relative to B.[ple...

Question

Let B be the basis of Pa consisting of the Hermite polynomials2+412 and 12t + 8t3 _ and let P(t) = 1+ 412 _ 8t? . Find the coordinate vector of _ relative to B.[ple

Let B be the basis of Pa consisting of the Hermite polynomials 2+412 and 12t + 8t3 _ and let P(t) = 1+ 412 _ 8t? . Find the coordinate vector of _ relative to B. [ple



Answers

Let $\mathcal{B}$ be the basis of $\mathbb{P}_{2}$ consisting of the first three Laguerre polynomials listed in Exercise $22,$ and let $\mathbf{p}(t)=7-8 t+3 t^{2} .$ Find the coordinate vector of $\mathbf{p}$ relative to $\mathcal{B}$

Yeah, in this trouble were given a set up. So we'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'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'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'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.

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'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.

In this problem, we'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'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'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't see too to be negative three and CP to be negative too. Said so it on the coordinates back to that we're looking for is seven negative beat and negative too.

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's square and one plus two tea. Okay, so the standard days, that'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's so here'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's the system we need to solve. So we apply the gushing in the nation two. Why don't we? I'll just write the result here. That is 100 They'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'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'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.


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