let-p_0-p_1-and-p_2-be-the-orthogonal-polynomials-described-in-example-5-where-the-inner-product-on-

Name: let p_0 p_1 and p_2 be the orthogonal polynomials described in example 5 where the inner product on
Uploaded: 2020-12-16
Duration: 8 min 40 s
Channel: Numerade
Description: Let $p_{0}, p_{1},$ and $p_{2}$ be the orthogonal polynomials described in Example $5,$ where the inner product on $\mathbb{P}_{4}$ is given by evaluation at $-2,-1,0,1,$ and $2 .$ Find the orthogonal projection of $t^{3}$ onto $\operatorname{Span}\left\{p_{0}, p_{1}, p_{2}\right\}$

Question

Let $p_{0}, p_{1},$ and $p_{2}$ be the orthogonal polynomials described in Example $5,$ where the inner product on $\mathbb{P}_{4}$ is given by evaluation at $-2,-1,0,1,$ and $2 .$ Find the orthogonal projection of $t^{3}$ onto $\operatorname{Span}\left\{p_{0}, p_{1}, p_{2}\right\}$

Anthony Ramos · Accepted Answer

Yes. Okay, so for this exercise, you got these three protectors point on. No. Uh, here. Okay, so this point on your speed, p zero b one b two are on B four. Okay. Aren&#x27;t the evaluations, um, are defined as zero equals two minus two to one minus one 2 to 0 231 g four to. Okay, so these are the evaluations off the polynomial for that we&#x27;re going to use to compute the inner product. And they remember that in this case, for arbitrary P and Q Yeah. Be cute. Elements off before the inner product is defined as a summation or from I equals to zero toe four or be evaluated. A the I times. Cute. Evaluated a t. I. Okay, so this is the way off. How? It&#x27;s fine here. The inner product that is off. Okay. Onto the space. Okay, So let us define so left. Consider, love you of the spun off zero. Be one YouTube. Okay, on. We&#x27;re going to consider affect. Cute. Given by the cute on. We need to compute the production off. Cute on the subspace office on the space pine by p zero p one and p two. So this is a So this projection eyes define us the in their product off cube, with zero divided by the inner product off easier with itself. Times zero love to with the one you same claims to be one on the other victors. Pete too. Okay, so let us start competing these values here, these in their problems. So the inner product you think mhm Cuban zero is equal to to taking the inner product off the cube one. Okay, the values off the teeth are zero equals minus three minus one 2 to 0. It&#x27;s right here in three years to t three is one on T four is equal Stoute. So this in the product is going to be minus two. Que waas minus one cube. Last one, cube glass to cube. And this is just zero. Okay, so this term is not going to to a port, Anything to this projection. So this value just here. Let&#x27;s continue with this one. Things. Problems between Q on p one is just taking Thank you team on this When we&#x27;re going to evaluate, it is minus two to the fourth plus minus +14 plus +14 lost to the fore. This is just 34. And then let&#x27;s compute this inner product here. So cute on beat. So get this three. Cool square. Okay, minus two. Do you square minus two. And this is minus two to the three. Four minus two plus minus three minus one. Cooper one minus two plus one. Cute one plus one minus two. Um, blast to cube four minus two. Okay, so here this term council, with this term on this term council with this one. So this is just zero. So these other term also becomes zero. So we just need to focus on this time here. We have already compute this one. So we need to compute the inner product off the one with itself. So the inner product off the one with itself, he just taking the inner product off the with cheap. And this is minus two square plus minus one square, plus one squared. Plus two square on. This is supposed to tell. Okay, so at the end, the protection off the vector cute on the view. Is it close to 34 divided 10 t

Angelo Rendina · Answer

so, accent, Why they&#x27;re not the vectors containing the binomial speed and cue computer in minus 101 So, actually, the vector minus 40 in two. And why is the Vector 535? Now we&#x27;re asked to compute the projection on topi off cue, we call the definition of projection that Xena product off cue against B dividing in a product off be against B. And this number multiplies the vector p No, because of the definition of innit? Prague between Polly Normals. We know that didn&#x27;t probably between Q and P is why Durex and the inner product be against B is extra X and everything multiplies the bulletin. Opie. Now why the X we see is just minus Stan. An extra tax used 20 and his number multiplies be. And so all in all the protection off cue on Toby. He&#x27;s minus one house. Me

Angelo Rendina · Answer

we have the polynomial speed zero p one MP too, and they don&#x27;t know the constant function. One the degree. One problem lt in the degree to play multi square. We&#x27;re looking at this bully normals as vectors in the vector space off continues function from minus two, two plus two with dinner problem given by the integral off the product. So we&#x27;re we&#x27;re looking for are Fogell basis for the span off zero p one MP too. So we&#x27;re going to use the gram Schmidt the composition method. So let&#x27;s begin by defining a vector zero off the so this will be the first ah, element off the air for colonel basis. And let&#x27;s just put it equal to zero off team thought is the constant functional one. Let&#x27;s also compute dinner problem at zero against zero just because it will need later. So it is by definition of inner product, the integral from minus 2 to 2 off well at zero times of zero, so one times 11 So just in there of Juan de ti aunties, just uneasy computation is is for now. Let&#x27;s project the one until at zero. I remember that this is given by P one against a zero dividing at zero against zero, and this number multiplies the vector of zero of tea. Now I put one over four and that&#x27;s the ah in their product at zero against zero in the denominator. And now we have in the numerator p one against zero. So that&#x27;s the entire from minus 2 to 2 off just x the X net. I&#x27;m using X because I&#x27;m using t in another place. But remember, it is just Ah, dummy. Variable token is called the axe, and this number multiplies the function of zero T. But now X is a not function and we&#x27;re integrating over somatic domain. So disintegrate zero. And so these old projection is zero. Now we can proceed with the good screen, the composition. So we put F one off teen as everyone off the minus the projection off the one onto a zero. But now we&#x27;ve seen that the projection is just zero. So let&#x27;s just be one of tea. So the function t And again, let&#x27;s compute f one against a form because we need you later. Disease didn&#x27;t take it out from minus two 22 off the phone skirt, Softy square the tea and again, easy computation. 16 over three. And now, finally, that&#x27;s computer projection onto F zero off B two off t. So similarly, this is B two against zero. Dividing have zero against 00 f. Zero f t and disease 1/4 date. A girl minus 2 to 2 off X where the ex times have zero off t that did this 4/3. They&#x27;re multiplies at zero of tea, but it&#x27;s just because of function one. And now we complete the projection on life. One off, Petey off. T Saudis be two against one dividing F one against one. These old number that multiplies the vector F 1 50 disease three over 16 Didn&#x27;t a girl from minus 2 to 2 off B to F one. So that&#x27;s X cubed Jax. Everything multiplying the vector of fun of tea. But again, excuse. He&#x27;s on all the function on automatic domain. So these old entirely zero and therefore the projection is zero. And now we can complete the Gram Schmidt the composition because we put half to off t to be pizza off t minus the projection onto a zero off. He&#x27;s off t minus the projection onto F one off the tee off T and Disease T squared minus 4/3 nine minus zero. So the vectors f zero f one.

Angelo Rendina · Answer

we have your focus now. Basis zero p one Batou, given by the polynomial is won T and T Square Minister. We know these from example Five. So I mean, let me right zero bar for the vector obtained by computing P zero in minus two. Imara swan in zero in one and into So this is just the vector off ones and then be one bar is minus two, minus 10 and two and finally P to bar. He&#x27;s two minus one minus two minus one into. Now we write these vectors because we&#x27;re called the dinner plucked between fully normals. So in this case, for instance, PCR against zero he&#x27;s given by p zero bar don&#x27;t p zero bar. Sonny&#x27;s get is five, and similarly, we compute the one against the one to be 10 and p two against me, too, to be 14. We complete these numbers now because they&#x27;re going to use them now. We&#x27;ll soon when we&#x27;re doing the Grand Street process. So let&#x27;s go see there The Perimeter Cube, which is given by D Cubed. Let&#x27;s right, Q bar, of course, is minus eight minus 101 and eight and we want to apply the garnish meat process to Q to obtain well obtain up vector riches or for colonel to both be zero p one MP too. So now the computation is simplified by the fact that the vectors p zero be wanting me to our world vectors. The polynomial sze p zero we won&#x27;t be too are already or for Colonel. So all we need to do is to compute the projection off cue unwto p zero p one MP to Palestine would be zero, which is Q against zero divided by P 00 And of course, we recall that these is Q bar dot p zero bar divided by five the fire we computed previously and then P zeros of the constant function one and now you can compute a cube are 0.0 bars zero So this whole projection is the no vector. Then we compute the projection on Topi one off you seeming our lady this cure against the one divided by the one against the one you compute his number. It is the result to be 34 divided by 10 and then be wanted as the constant Well, the function t That&#x27;s what these 3.4 t and finally the projection onto P to off you He&#x27;s again Q against me, too divided by P two against me. To that multiplies the polynomial P too. And again you against see two turns out to be just zero. So this whole number is really bothered by 14. That multiplies the vector with very normal D squared minus two and he&#x27;s just zero. And therefore we can put the three to make you minus all these projections. And the fourth is too cute miners, while the only thing they were subtracting 3.40. Finally, we&#x27;re asked to scale this polynomial so that p three bar is minus 1202 and one. And of course, we see that in our particular case, be three bodies given my minus 1.220 months 2.4 and 1.2. So all we need to do is to scale this vector by one over 1.2, which is exactly 5/6. So if we divide by 1.2, we get the scale. Vector is carefully. Normal is 56 off a cube minus 17 6th off T. And in fact, this bully normal where we take the vector, Sit through it with the bar. He&#x27;s exactly minus 1 to 0 minus two and one has wanted.

Chris Trentman · Answer

you&#x27;re a and we&#x27;re also given characters of this matrix were asked Fine matrix s s these work Cardinal s your agonal eyes is a make sure say is negative 3 to 2 to native 32 to to negative three and characteristic equation we&#x27;re given is bland is one minus land ties land that plus five. It&#x27;s weird. In order to find this nature, it&#x27;s s person that Meters a is a riel symmetric matrix. So we know that a complete set of work, the normal vectors of a exists and if s is the matrix who&#x27;s calling that years? Are these Ortho normal? I bet yours then we will have by a fear of that&#x27;s matrix s is volatile. And that signal is is a in order to find this set, uh, or the normal vectors redefined the idea. Values said Mark characters equation. He had landed zero The team that the roots of the Kippers a polynomial which are the values are Linda. One equals negative five with a multiplicity off to and blended two bowls, one with a multiplicity of one. If x ones and high conductor associated with Lando one and satisfies the equation, a minus under one. Why x one equals zero. Becky Nature&#x27;s a minus like the one I is a Matrix a plus five i, which is tricks Negative. Three plus five to 222 222 Tubes of all twos Galaxy elimination. This becomes matrix 111 Heroes. You&#x27;re a hero. You&#x27;re 00 So we obtained this system Equations x 11 plus x +22 plus X +33 equals therefore X 11 is equal to negative X 22 My sex your two degrees of freedom. So take x two to be the perimeter are and x three This should be x 12 Next take x one to be the frame are X one the primer s Then I connect your X one is give our mind assess are yes This is equal to are times that after negative 11 year old Los best times A vector native 101 if you take are to be zero STD one we get that are vector X one is the back here Negative 101 Now because hiding space has multiplicity two young value 19 states has mentioned to so they want to find another I in vector of the second states, which is linearly dependent. So x two is the specter. You know, that satisfied the same equation as X one, and you know that it has to be or nearly so, yes. So now his ex two of the same expression with craters are nets. But ask people here, independent. It&#x27;s now this time let&#x27;s take backtracks to be when our Z one sec and then we get that the vector are is backtracks to is negative 110 and clearly this is nearly attendant from X one, the end product off X one and x two. We simply don&#x27;t crafted excellent axe to It&#x27;s the Real Numbers, which is one plus zero waas zero, which is one so x one x two are not or Taliban. Find to iron makers of space at work. Father, we&#x27;ll have to use the Grand Schmidt process. So let V one you get that here x one and let the to be back here next to minus projection of X to on to be one, which is the aircraft off X two. You want over a normal one. It&#x27;s where times a vector the one. And he calculated the air product of X two and next one, which is the same as their card. Next to view one, this will be next to minus one, and the normal one squared is in all of X one squared. No, the next once were going to be equal to native one square, one plus one square one. This is too. So you two x two minus 1/2 do you want? And this is going to be the vector Negative one minus 1/2 negative one or negative one plus 1/2 or 1/2 one minus 1/2 0 or one and zero. I guess we have one or negative one. And so we have now to Decker&#x27;s Do you want to be, too, which are worth talking to each other by Graham Shit Normal that there&#x27;s normalize 12 So, in effect, you want this is do you wanna over the normal be one which is calculated in one is square root too. And the norm, um be to is equal to, I swear, while negative half squared, which is 14 plus once where one. Let&#x27;s negative past where one for or three halfs. This is the square root. Three or two. We have another vector you want? Yes, Native one of her too. Zero one of her too. And after you too. It&#x27;s the unit that here. Negative 1/2 divided. 5432 or negative 1/2. We&#x27;re to over three. Oh, she does accurately route 2/3 and make it will have. We&#x27;re 2/3. This could be simplified, but for now, in this form that actually go. So what happened by my recovery? Get negative. Six over two times 36 six Over three. I began making six. And finally, yeah. Next three is the conductor associative that I thought you wanted to And ex trees weren&#x27;t satisfied. The equation a minus landed too. I x three people to his euro vector. And we have that The matrix minus landed to I is the matrix a minus I, which is being negative 42 to make it for 2 to 4 with fallacy and elimination, New tricks becomes against things around. Inviting by never grow by two you get 112 one maybe two, one and they get to 11 Her gassing combination adding tracking the first From the second you get 11 day too zero negative minus three one. My snake it to his three and adding twice the first rate the last year we get zero one plus Jews three one us negative for negative three and with more elimination becomes one and then dividing the second row by three. After already admits Arthur grow will be 000 our second rose. Now you&#x27;re a negative one one. And we was your one negative forms. Got negative. Three. Who&#x27;s attract second row from the first row to the rose. 01 negative one from road 112 We get one one minus 10 And you too, my ass Native one is negative or second load. Name one. This new tricks murders form. See this system X 31 minus X 33 equals zero and x 32 minus 63 equals zero Senate. Actually one called 63 and to equals x victory. So we take extra three to be parameter are then the I connect Your x three is effective are are are because the same is the next year Our times 111 We take our to be one of my vector x three. This is about your 111 This sector is already or thoughtful too, and linearly independent from the other two vectors. However, we need to normalize it. Norm Next three is the swearing one square, this one straight. This one squared plus one this one or three and therefore unit after you three is one of her three. Whatever three one of. And so, as I said earlier, you hear me says that you choose. That&#x27;s to be matrix home. That yours. Are you on your 23? So Mega one of her 20 want to you call it is negative. Negative 6/6 6736 six. Final call is one of her three. One of every three says theorem tells us that sneaky tricks is or probable, Jeanne, that&#x27;s universe is equal to its transpose. So that s trans was a s Will be to Azinger&#x27;s a s. And because the column that there&#x27;s best are also linearly I collect use a yes, I analyzes a So this is equal to the diagonal matrix. These entries are ivy values a which we determined were negative five C two and one. Therefore, this matrix s is a matrix that we&#x27;re looking for

Ashley Boni · Answer

So first we need to check that you wanted me to or thought Gotham by taking dot product So you won. Uh, not You too. So we get zero minus one plus one and that zero So those are welcoming each other. So now we need to find the doctor. Call it white hat. Um, what is the orthogonal projection of? Why? Onto the span of you wanted me to. So we find this by projecting Why onto your bomb. So you do that by taking dot product of the two over dot product of you one with itself and not apply. Buy you one and add to that the same thing. But projecting onto you too. Eso Let&#x27;s compute all the dot products off to the side. Why dot You won gives us minus 24 minus four. Close one. So negative 28 plus warm negative 27. You won dot product with itself a 16 close one plus one. So we&#x27;ll get 18. Why dot product with you too get zero plus four plus one. So five and finally you to doubt product with itself. Zero plus one plus one. So we get to so our first coefficient is negative. 27 or her 18 which we can simplify two minus three halfs. And 2nd 1 you get five over, Chip. So now we just want to multiply these coefficients through to the vectors s o negative. Three half&#x27;s times you won give us 12 over to just six. And then they gave 3/2 times negative. One three halfs. Negative. 3/2 times, one minus that halves and then multiply it by five laps. And to you, too, you get a zero moon five halfs, five halfs. Adding these together 606 three halves and 5/2 says eight halves before and 5/8 minus three. Half says two halfs more s. So that is our director weren&#x27;t interested in

Problem

Find a polynomial $p_{3}$ such that $\left\{p_{0}…

Problem 11 Easy Difficulty

Let $p_{0}, p_{1},$ and $p_{2}$ be the orthogonal polynomials described in Example $5,$ where the inner product on $\mathbb{P}_{4}$ is given by evaluation at $-2,-1,0,1,$ and $2 .$ Find the orthogonal projection of $t^{3}$ onto $\operatorname{Span}\left\{p_{0}, p_{1}, p_{2}\right\}$

Answer

see answer

Related Courses

Linear Algebra and Its Applications

Related Topics

Discussion

Top Calculus 3 Educators

Catherine R.

Heather Z.

Caleb E.

Joseph L.

Calculus 3 Courses

Vectors Intro

Vector Basics - Example 1

Recommended Videos

Watch More Solved Questions in Chapter 6

Video Transcript

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

Linear Algebra and Its Applications

Catherine R.

Heather Z.

Caleb E.

Joseph L.

Vectors Intro

Vector Basics - Example 1

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

see answer

Linear Algebra and Its Applications

Catherine R.

Heather Z.

Caleb E.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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

Catherine R.

Heather Z.

Caleb E.

Joseph L.

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

Linear Algebra and Its Applications