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Polnts) Let Pn be the vector space all polynomials degree or less the variable Let D : Pz operator LetPz be the Iinear transtormation derlned by D(p(z)) # (x): That...

Question

Polnts) Let Pn be the vector space all polynomials degree or less the variable Let D : Pz operator LetPz be the Iinear transtormation derlned by D(p(z)) # (x): That Dis the derivative{1,I,22,2*} {1,T,12},be ordered bases for Pa and Pz, respectively: Find the matrix [DIG for D relative the basis B in the domain and €C in the codomain[DIS

polnts) Let Pn be the vector space all polynomials degree or less the variable Let D : Pz operator Let Pz be the Iinear transtormation derlned by D(p(z)) # (x): That Dis the derivative {1,I,22,2*} {1,T,12}, be ordered bases for Pa and Pz, respectively: Find the matrix [DIG for D relative the basis B in the domain and €C in the codomain [DIS



Answers

(Calculus required) Let $D: P_{2} \rightarrow P_{2}$ be the differentiation operator $D(\mathbf{p})=p^{\prime}(x)$ (a) Find the matrix for $D$ relative to the basis $B=\left\{\mathbf{p}_{1}, \mathbf{p}_{2}, \mathbf{p}_{3}\right\}$ for $P_{2}$ in which $\mathbf{p}_{1}=1, \mathbf{p}_{2}=x, \mathbf{p}_{3}=x^{2}$ (b) Use the matrix in part (a) to compute $D\left(6-6 x+24 x^{2}\right)$

Hello there. Okay, so for this exercise we have the following linear operator that actually corresponds to the differentiation operator. And it's fine just by taking a polynomial in the space of animals of music too. And it returns it's directed as simple as that. However we need to find here the matrix representation for this linear operator in the basis B. And the basis B is defined by the following polynomial two to minus three X. And two minus three X plus eight X square. So we know that in other to construct our metrics, we need to apply the differentiation operator to each element in the basis. So let us start with two. The differentiation of two. Well is just zero. The derivative of these. The other Elements in the basis that is 2 -3 x. Is equal to -3. Unlike lastly this third element in the basis is the derivative of two minus three. X was eight x square. This will be the victor. The polynomial. 16 x minus three. No, these are the points are meals But we need to represent this in this base. The basis. B. For that We need to write well this is clear it's going to be the zero vector But -3 is not necessarily the zero vector. The formal procedure as well. We're going to create this in the standard basis of paranormal. That means 300. And we need to find the coefficients That satisfied that Alpha one times a vector. 200. These elements bases plus alpha two 2 -30. That is these element bases Plus all for three. The disease was 2- -3.8. Okay. This this german basis. We need to find the coefficients that satisfy this equation in this case That corresponds to Alpha one equals 2 -3/2. Alfa two equals to zero and alpha three equals to zero. Okay, I'm sorry. Here is -3. Whether to avoid country. So the representation of -3 in the space B is the vector -3 helps zero and zero. The same. We need to do for this polynomial and we will find that the vector in the basis B Is defined by -3. Bye. 23/6 And -16/3. See it this Of course in the basis. Now with all these vectors find in the basis be we can construct or the or representation for this transformation. So that means we need to put each vector as a column of the matrix. So the matrix or this differentiation operator in the basis B is the final first. By the zero vector then by the better minus three helps 00. And here 23 six -16 Divided three, zero. And there's make things now we need to play this. Right? So we need to find the derivative of the vector of the meat of the polynomial. Six -6 x 24 x square. We need to do this by two step procedure. Three step procedure. So in the first step we need to transform the paranormal into the basis that we want. In this case is the basis beat. And we know that the basis B is defined by the following polynomial. Two minus three X plus eight X. S square. So this point represented in this basis corresponds to the vector one -1 and three. That's the first step. In the second step we need to multiply the matrix that represent this transformation by the picture. This case we're going to call X in the basis B. And this will return as the transformation of this vector or in this case polynomial. But in the basis B. So let's speak the matrix for these transformation was 000 -3 Health. 00. And the last one column was 23 divided six and 13 -16, divide three. Here time is a vector That represents the polynomial in this basis. So 1 -1, 3. And the result of this matrix multiplication is a vector. 13 -16 0. In the basis beep. Okay, but that isn't the basis be. We don't want the result in the basis be we want in the standard basis for the space of polynomial. So we need to Kind of the transformative or change of basis to be more precise. So we have here the 13 -16 0 in the basis beat. So in order to transform it, we just need to multiply 13 Times two which is the basis element of B minus 16. And the other basis element That is 2 -3 x. The result of this is just -6 plus 48 x. Okay. And in case that you want to check the result, what you kind of fly directly the transformation. In this case we need to verify the full with polynomial Plus 24 square and you can observe that indeed the derivative of this polynomial is just -6 plus for eight x. That is just what we obtained here. Following the three step procedure. Of course.

Work. Given the faces, that is, um, one one minus t. And to minus 40 us t squared. Yeah. So we have these three a nominal size. Our faces. So we honored to Right down. He up t in terms off. Coordinated back. Factor off P lifted two spaces. First down, Petey is seven miners on a safety plus re t squared. Get less. Dean owed one as p one and one minus TSP too. And do you mind us? 40 plus t squared as p three. So in order to get three t squared, the only the only way to get people can t squared is too is by using the p three. So at least we have to times p three by three. That way we can get three t squared minus well, t, that's sick. Only in this way we can get pretty squared. Otherwise, we don't have t squared in p to m p one. So And you too. We need to combine Pete even p three to get connective and activate e. And we What we have for Peter is negative t and we've already had we Robert He already had a negative 12 so the only thing we need to do is two times you to buy electric port so that he so that this this, uh, this is equal to negative for last 40 or teeth. Okay. So that our, uh, Osama off three p three milers, four p two will be well, give us negative 80. So, in terms off and then ask in terms off a constant term, we have one. But previous previous stabs our already give us, uh, to because six minus four would be too s o. The only thing we need to add is fight so that to minus to plus five will be seven, which is here. So our final answer will be first three p three minus 42 plus p one. Well, five, if you want. All right, So this is our by no answer.

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.


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