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1. Given a basis B = {1 + 1,2 - € +22 1 = 22} of Pz; find the coordinate vector [2 + 225; find the polynomial p(z) € Pz with the coordinates[p(z)ls...

Question

1. Given a basis B = {1 + 1,2 - € +22 1 = 22} of Pz; find the coordinate vector [2 + 225; find the polynomial p(z) € Pz with the coordinates[p(z)ls

1. Given a basis B = {1 + 1,2 - € +22 1 = 22} of Pz; find the coordinate vector [2 + 225; find the polynomial p(z) € Pz with the coordinates [p(z)ls



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 $\mathcal{B}$

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.

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, we're going to create vector V and it has an initial and terminal point. So for each of our components we're going to take our terminal minus our initial. So our vector V is going to be two minus six comma six minus two, comma two minus two. So that gives us a vector of negative four comma positive for common zero. So when we write that in, our component form will have a negative for I plus for jay. And because there's a zero in that K piece, we do not have to write it.

Okay, we're going to be fine in vector V. That has initial point P one in a terminal point P. Two. So in each case we're gonna be taking our P two minus R. P one to find our component. So our vector will be a three minus six comma zero minus two, comma two minus one. That gives us a vector of negative three comma negative two comma one. How to write that in. Its component form will be a negative three I plus two J. And then we can just say plus K. You don't have to say plus one K. By saying plus K. It assumes the value one.


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