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2.6.10) Find the standard basis of functions associated with the characteristic polynomial as) = 82 + 282.6.14) Find the standard basis of functions associated with...

Question

2.6.10) Find the standard basis of functions associated with the characteristic polynomial as) = 82 + 282.6.14) Find the standard basis of functions associated with the characteristic polynomial 7s) 82 + 9 _2.7.24) Find the standard basis of functions associated with the characteristic polynomial a(s) 83 + 8.

2.6.10) Find the standard basis of functions associated with the characteristic polynomial as) = 82 + 28 2.6.14) Find the standard basis of functions associated with the characteristic polynomial 7s) 82 + 9 _ 2.7.24) Find the standard basis of functions associated with the characteristic polynomial a(s) 83 + 8.



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 $\mathcal{B}$

The function traffics is given to us, which is equal to 2.4 x cubed with the stool point for bless five off Excuse five off X square. Other. This is X squared minus X less seven. Worried now leading Tom. This is equal to the term containing the highest rubber. Now we can simplify this also like we can take the CME so it will be multiplied toe this. But rather than doing so, we can simply answer the leading term. Okay, because after doing that, also we have to divide everything by. So the 2.4 will ultimately come. So leading time will be the term associated. But I guess for which is 2.4 excuse leading coefficient Leading coefficient is the coefficient of the leading damages to point for their degree off. The situation is three. Now we have to decide that video natives who are ready quickly nearby quadratic cubic. Since the U. S. Travelers three. So this is cubic commission

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.

This is a problem. Number 73 We need to find the standard A question of parabola given according to the given graph the parabola eyes. Okay, this is three comma one which is our vertex is three comma one and the parable is passing through to common zero and for common zero toe comma zero. And for Camacho that these are the acts intercept. Okay, So since parabola is downward parabola so the standard equation will be written as X minus that whole square equal toe for P Why minus k where HK is the Vertex sk is given as three comma one. So let us plug in the values of X and K X minus three whole square equal toe for be Why minus one. Now this parable passes through two comma zero So let us plug in to comma zero that is executed to and why I could zero in this equation to get value for P So tu minus three old square equal to four p zero minus one So two miles 11 equal toe four p with negative sign. So four p equal to minus one. This is our four p. So let us plug in any question One, we will be getting required. A question is X minus three old square, equal to minus one. And my son Y minus one. So x minus three or square tickle toe minus. Why? Minus one. So this is the standard question. Thank you.

This is 74 number. Question The given parabola is like this, and it has Vertex five common three and it passes through 4.5. Come on. Four or it also passes through zero common zero. Okay, not so it is confirmed. This point is confirmed. So will be writing 4.5 common for this passing through this point. What Exit five. Common three. So X comma, Magical toe five, comma three. And this isn't a leftward opening parabola. So a question will be why minus k Old Squire equal to four p x minus set. So let's plug in. The values of HNK at is five k three. So why minus three? Holds quite equal to four p x minus five. So we have this is passing through this parable passing the four point 4.5 comma four, Which means we need to plug in the values 4.5 and executed. Opened five. Why? I called for in question number one. Let us say four minus three holds quite equal to four p 4.5 minus four. I'm sorry for 15 minus five. So this is one equal to four p into minus 0.5, that is, this'll become for a negative off to pee. Okay, but we need to find the value of four p. So four p definitely will be equal to minus two. So let us plug in in question number one. Why? Minus three or square will be minus two X minus five. Look, So this is Dancer. Thank you so much.


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