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5. Let D be the derivative operator on 4 = {ao + a1x+a2r2a; € #} Find the matrix which represents D relative to the basis...

Question

5. Let D be the derivative operator on 4 = {ao + a1x+a2r2a; € #} Find the matrix which represents D relative to the basis

5. Let D be the derivative operator on 4 = {ao + a1x+a2r2a; € #} Find the matrix which represents D relative to the basis



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.

Okay, so now we have here, the differential operator that is defined in the space of real valued functions in the interval minus infinity, infinity. We need to find the matrix representation for this transformation in the basis B. The basis B is composed by one sine of X. And co sign off X. Okay, so basically if we want to transfer to obtain these matrix, we need to apply these differential operator to each element of the basis. So here Derivative of one. Well, this clearly zero. The derivative of the sine of X is the co sign of X. And finally the derivative of deco sine of X is just minus nine X. Okay, so we need to write this in the basis B. So technically zero corresponds to the zero factor in this basis being The co sign while this corresponds to the vector 001 in the basis B. And finally the minus sign Is represented by 0 -1, 0 in the basis. Then we need to pick all these pictures to construct our matrix. So the matrix of this differential operator in the space, in relative to the basis B Is defined by the Matrix 000 001 And 0910. These are zeros rate. And as a part of an application going to right here again, the matrix Mine is 10. So now for the part B we need to calculate either relative of the following function two plus three, sign x minus, or calls on Yes. Okay, so how to do this? Well, we have a three step procedure, the first step is transforming dysfunction into the basis. B. So two plus three sign X minus for co sign X corresponds to the vector in the basis B. 2 3 -4. Then in the second step we need to multiply our matrix by the vector that we obtain 2, -4 in the basis. Be in order to obtain the transformation. Okay, so here we have the matrix 000001 0 -10. Applied to the vector 2, -4. And the result of this is the picture zero for and three. This is in the basis B. And the last step for this transformation to train. The transformation is just right this E as a function. So this represents actually four times the sine of X. Uh huh plus three times the coastline of X. And there's a result. And actually you can verify this because what the exercise is asking is to find the relative of the following function. two, Sine X -4 Co Sign X. And wealthy derivative of this function is three. Co sign X plus four. Sine X. That is what we have here


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