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Consider the inner product space Pa(R) with inner product{p(c) 42)} EekrJqkr) dx: points) Apply the Grau-- Schmidt process Lo the basis+ rof the subspace Pz(R) (le...

Question

Consider the inner product space Pa(R) with inner product{p(c) 42)} EekrJqkr) dx: points) Apply the Grau-- Schmidt process Lo the basis+ rof the subspace Pz(R) (leave answers in exaet form)(6) (2 points) Find the polynomial in Pz(R) which best approximates 212 to the given inner product_ (leave answer exact form)with respect

Consider the inner product space Pa(R) with inner product {p(c) 42)} EekrJqkr) dx: points) Apply the Grau-- Schmidt process Lo the basis + r of the subspace Pz(R) (leave answers in exaet form) (6) (2 points) Find the polynomial in Pz(R) which best approximates 212 to the given inner product_ (leave answer exact form) with respect



Answers

Let $V$ be the space $C[-2,2]$ with the inner product of Example $7 .$ Find an orthogonal basis for the subspace spanned by the polynomials $1, t,$ and $t^{2} .$

Hello there. So for this exercise we start with a subspace that is respond by these three vectors. We want B. two and b. three. Okay. And we need to find an Ortho normal basis for the space. So the first step is to, well the point is that we know a procedure that takes a set factors. Yeah two mm. And then it returns after playing the punishment procedure. It returns a set of orphan yeah sets the Northern Normal said. In terms also. So you start with a set of pictures and you obtain an Ortho normal set. But to apply this procedure we need to know that these vectors are linearly. Mhm. So before starting to calculate and or applying the garnishment procedure for this set of B one, B two and B. Three. What we need to do is let's remind if these vectors are Yeah. And to do that we need to check if one of them is reading as a combination of the other one. And you can observe that actually The Rector B three Is equal to be one. Yes. Beach. So you can observe that this just by inspection if you want a procedure to check this. So process or agree them to do that. What what I recommend to do is put in these factors in a matrix form. That means putting the vectors as Rolls of our matrix. That 012 -101 -113. Okay. And here then you try to reduce this matrix to the action form by applying the girls procedure and here you will observe that when you want to eliminate this. Mhm Put a zero here or here you will obtain at the end you only two pilots one and hear something like the A. B. Here. C. 00 Yeah that means that one of your rose will become full of zeros. And that means that you is enough to pick you. Only two of the vectors that you're considering this in this sense, If you obtain three pilots after applying the girls elimination Using three pilots, then your three factors are linear. But in this case what happened is you obtain something of this form or you can check based by Inspection that the effect to be three is a linear combination of the. Okay, so first you that your mind that if the set is linear independent and we observed that it is not. So it's enough to eliminate one of the factors that you have here This case I'm going to be three and we're going to use we one and two. So for this victory people for this space is enough. two. Great. That is the span of the factor we want and beat and they said expand the subspace up and they are linear independence. So now we can apply the garnishment procedure to obtain a basis. That is all for now. Okay, so let's remember that for the grant smith procedure we are going to obtain, We're going to pick this had to be one B two. And after applying the Greenwich you're gift in a set of factors Of the one Alpha 2. Both unitary vectors. To do that. We need to Fix one of the vectors here, say alpha one here. I'm used the Yeah. For unitary factor. So this vector here is not unitary. So we fixed one of the batteries in our set Either be one of you to to follow the same order. I'm going to be one. So alpha one. It's going to be B one. That means 012 Is the 1st step. So in or set. In our final set, we have already director of what Then the next step is the second vector offer to Such that this Alpha two is a phony one. And to do that we need to take the vector V two and transform Director of Alpha two Such that this vector is also an out for one. To do that we need to right offer to us the vector V two minus the projection of YouTube on all. For one When we do this we are obstructing the projection. We're eliminating the component of YouTube that is aligned with alpha one and the result will be and Better. That is Arthur 12 for one. So in this case This becomes the Alpha two is equal, two minus one, one minus. And here the projection just to remind you that the formula for the projection of a vector into another one is equal to you. The inner product of these two vectors times that picture to the one that we're projecting, two divided the norm of this square, enormous. So this projection particular case is equal to -250 times 012 And this part you can observe that corresponds to awful. So after subtracting these two pictures, We obtained that Alpha two is equal two minus one minus 2/5 And 1/15. Okay, the second vector orthogonal set. So what we have so far is a set of orthogonal factors Okay over one Equal to 0 1, two. Alfa two equal to -1 -2 and one over deep. But what we need is an Ortho normal set. So so far this is or a phone but we need say that should be normal. That means that the factors are unitary. And to do that we need just to normalize the vectors. That means that alpha one, you need to hurry Will be all for one divided the normal album. This is just One over the square root of 501 two. And for alpha tube we applied the same formula. So we normalize the vector by divider but it's not and we obtain one over The square root of 30 times minus five minus 21 These two vectors for or phone normals. Ah these vectors, so all for one, All for two, the span of these vectors, he is a subspace of you even more. We know that by theorem 6.3.1 Ortho normal set is linear. And then so that means that these two vectors are linearly independent. Therefore we call director. We will be Formed by Alpha one oh two. Is a basis is an is an orphan, success is an old on our mountain basis for done

Were given a subspace W. Which is the set of all polynomial of degree at most. two P. Two with an inner product inner product of F. And G is the integral from 0 to 1 of fft times. G M T E T. So oh yeah, interested in the projection of the function F f T equals T cute onto the space. W. As a hint, we're told you the orthogonal pollen or meals one To T -1 -1 and 60 squared minus 60 plus one. And then you can So we found that these were orthogonal in exercise 722. Yeah. Now because these are our dog kennel, we can just calculate the fourier coefficients to find a projection of that's under the space. Yeah, these are an orthogonal basis. Mm mm for W. Okay. Yeah. So before a coefficient C one is in a product of t cubed with one Over the inner product of one with itself and Yes. Right. Yeah. So this is the integral from 0 to 1 of t cubed D. T Over the integral from 0 to 1 of one gt. This is uh 1/4 Over one which is 1/4. Yeah the second fourier coefficients. C two is the inner product of t cubed with two t minus one Over the inner product of two T -1 with itself. This is the integral from 0 to 1 of well, t cubed times to t minus one DT over. The integral from 0-1 of to t minus one squared E T. This is equal to the integral from 0 to 1 of two T to the fourth minus t cube pt over the integral from 0 to 1 of four, T squared plus minus 40 plus one B two. Right. And this is equal to Take me up to derivatives and evaluating to 5th drugs. 14 over four thirds minus two plus one. This is to Fix. -14 is 8 20th -5, 20 assists 3 20th We were 4/3 -2 4/3 6/3 is negative. 2/3 plus one plus three thirds is positive. One third which is mm 9/20 right. You know, I like like in the field and finally for a coefficient C3 this is the inner product of she accused, I will be with uh then, I mean New Orleans, these Chicago, the 21st and 22nd, 60 squared -60 plus one. Yeah, pleased by the over the inner product of 60 squared minus 60 plus one with itself. What he said This is the integral from 0 to 1 of t cubed times 60 squared minus 60 plus one. Bt over the integral from 0 to 1 of 60 squared -60 plus one squared E. T. Yeah back I mean mhm. The this is the integral from 0 to 1 of uh 60 to the fifth minus 62. The fourth plus T cubed E. T. over. The integral from 0 to 1 of this is a little tougher. Doctor said 36 T. to the 4th. Mhm. 36 T. to the 4th -72 T Cubed. I do act plus 48 T squared minus 12 T. Mhm. No it's somebody that cares. Plus one. Well he's not taking anti derivative and evaluating we get one minus 6/5 plus 1/4. Yeah. Yeah. Mhm. 36 5th minus. Yeah. Yes. Thinking of buddy 18. Mhm. Yeah. Plus uh 16 minus six plus one. This is uh 40th minus 24/20 is negative. 4 20th Plus 5 20th says positive. 1 20 over. Uh six in Melbourne, Melbourne that 36/5 95th. Yeah, This is 1 5th he looked Which is 5 20th so or 1/4. Mhm. And therefore, oh okay, projection of our function F onto our space W This is going to be C1 times are first function one Plus C, two times their second function to T -1 Plus C. three times our third function 60 squared minus 60 plus one. So, plugging in this is uh 1/4 plus 9/20 Times to T -1 plus 1/4 times 60 squared minus 60 plus one. Mhm. This sympathize too. Three halves, t squared minus 3/5 T plus 1/20.

Mario is. Yeah. He's just trying to see some. Are you were given the vector space V. Of polynomial over our Of degree less than or equal to two. I tried to like I can't I cannot tell you one. I have no idea what happens he said to to you here. No, I don't. Dude samba music. You refer over our with an inner product. Find by the inner product of two functions. F. M. G. Is the integral from 0 to 1 of the 50 chinese G. O T B. T. Website. Whereas defend the basis of the suspects. W. Orthogonal to the function ancient too. Did you see it was to T. Plus one beaches. Dude rob thomas number one song. Well that's the that's the song. We should. First of all We know that affects here. I'll call F. zero will be orthogonal to each. If the inner product of that zero and H equals 01 other words zero is equal to he simply a final. Yeah. Any time we have to google swell show really? I'll let F. Zero T. V. Equal to 80 squared plus B. T. Plus C. And therefore the inner product of S. Zero T. Do a speech. You are sneak. Is any girl from 0 to 1 of a. T squared plus D. T. Plus E. Times. Yeah To T. Plus one. Security. Yeah. I went orgy was you want this to be equal to zero? Well let's solve this integral. Mhm. Thanks. Come on. This is this game shit Adams this is the integral from here or the one uh to a T. Cube. It was because plus en plus two B. T. Squared. Plus he plus you've seen how risky pussy time be thine. Yes. Well this is equal to getting the derivatives. Uh A. T. To the fourth over to bless State was to be over three and 2 kids plus B. Plus to see over +22 squared plus two times two continue called 0-1 which is over two plus A plus to be over three plus B. Plus to see over to plus C. And this is a favor to plus A with the room. The idea of which is yes. 56 oh plus uh Seventh shit. E. To see you then want us to be equal to zero. Let's speak now for example what one king goo one. Yeah uh refuse on earth knows what C. equal to zero and take be equal to ah negative five. Then it follows a. Is equal to uh positive seven. And so we get that that zero of tea is 70 squared minus five to how's your by Giannis? Because that shit looks like an Iranian Russians. Now we know that there's one more victory. No thesis. But it's so spruce To find it. You know that this vector f. one of T. Which are called A. T. Squared plus B. T. Plus C. Well it should just be port. Mhm. And what's your we want the inner product of once again F. Zero and H. Zero. So once again we want 56 A. Aliens get home washing plus seven six B. Plus. To see equal zero. We also want in a product of F. One. H. T. V. Zero To be in a project F. one, mate. Well sorry that's okay. This is they're rubens were records. It's the person that's what actually instead to find another vector. First four star general to H. And also nearly independent the F. Two or F. Zero. Well he's good. Let's take uh These are the zero and let's take mhm. A. to b. 12. And if I was that C equals -5. Have since equals negative five In F zero and equals zero and Uh one opposed F one and a 200. The independence FT is given by 12 to swear minus me minus five. As have you ever been, There are always like and therefore F zero and F. one. Yeah. From a basis for or some space. Help me.

So here our if one eggs is he code Teoh one past two x and have to It's ICO to minus two minus X squared. So a review one the previous crimps meats V one is because you're F one. It's 1002 x and I will beat you Now be have to minus that, you know production between have to and the one is a productive to B one and B one and B one. So first we computes a unit protects it. That's after a and B one to advance his ICO to their years ago. From 0 to 1 one plus two x minus to plus minus X plus X squared DX. So around here it's one is the A minus two minus buying friends are so that's expended house So minus two Money's floor X learning six minus two X Square. That's X Square, thus to excuse the gigs, Then you get minus two minus phi X minus. X squared does excuse. So they say coaching minus two minus by over two minus 1/3 Does two or four. So this number 12 because my money's worth to money for 312 My 13 Overstreet. And we also computed everyone. Be one. You know, protect here will be 0 to 11 past two x one of two eggs. Yes. So this will you come to nearer to 11 four x boys Where and speaking to one house over to us over ST this will too. 34. So me, too, is Nico too. Minus two minus X plus the square and miners mining. Certainly over, sweetie. 30/3 in the one one plus two x and this will be minus two minus X plus X clear, plus one plus two x So this will be minus one x squared. The the ground war. New York It's the basis it's equal to one that's to eggs in another one will be minus one house X square.


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