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Suppose that B = {V1, V2, V3} a basis for R? whereV13)' V2-3 -2and VgLet vbe a vector in R? -2Find the coordinate vector of v with respect to B. You may find t...

Question

Suppose that B = {V1, V2, V3} a basis for R? whereV13)' V2-3 -2and VgLet vbe a vector in R? -2Find the coordinate vector of v with respect to B. You may find the following reduction useful.5 =9 4 2 3 28 2The matrix-3 -3 -2~4by Maple, is reduced to-2Enter your answer in the box below; in Maple syntax: [vJBNote: The vectorin Maple syntax; should be entered as <a,b,c>

Suppose that B = {V1, V2, V3} a basis for R? where V1 3)' V2 -3 -2 and Vg Let v be a vector in R? -2 Find the coordinate vector of v with respect to B. You may find the following reduction useful. 5 =9 4 2 3 28 2 The matrix -3 -3 -2 ~4 by Maple, is reduced to -2 Enter your answer in the box below; in Maple syntax: [vJB Note: The vector in Maple syntax; should be entered as <a,b,c>



Answers

Let $B=\left\{v_{1}, v_{2}, v_{3}, v_{4}\right\}$ be a basis for a vector space $V$. Find the matrix with respect to $B$ for the linear operator $T: V \rightarrow V$ defined by $T\left(\mathbf{v}_{1}\right)=\mathbf{v}_{2}, T\left(\mathbf{v}_{2}\right)=\mathbf{v}_{3}, T\left(\mathbf{v}_{3}\right)=\mathbf{v}_{4}, T\left(\mathbf{v}_{4}\right)=\mathbf{v}_{1}$

Hello there. Okay. So for this exercise, what we need used to mainly transformation matrix relative to the basis the prime. And be That are defined here. So the basis, the corresponding vectors 1, 3 and -4 and the basis the prime are formed by these three vectors B one Is 1,112,220. And between 300. The transformation that is applying this case is the finest follows you pick our to lecture and you return and our throwback in this for Okay. So how to find this transformation. The first thing that we need to do is will apply the transformation to each of the elements in the basis of the domain. In this case the biggest part of the domain. So we need to play first transform meat sector. You want Okay when we apply the confirmation to this vector we obtain the following result. So supply days we have seen defectors in our three 11 minus 10. Okay but then we need to regret this vector in the basis of okay. Of the product means how it's going to be right. This vector in the basis of the prime. Right. How to do this. Well basically we just need to find three quick efficient over one Over two and 3. two. Great. This vector as a linear combination of the basic elements of the crime. 111 here you 20. I'm here is 30. Okay so in this first part the coefficient are So all for one Is the cost to zero of the two equals 2 minus one Hull. And also three Is equal to eight. So that means that these pictures all reading better. T of you want in the base of the prime corresponds to the picture. 0 -1/2 and 8:30. Now we need to repeat this procedure for 30 seconds. Mm. So he applied to Youtube. Yeah, they're in the usual rules of the transformation. We obtain the vector think 20. And that when we transform this into the basic elements of the prime, we obtained that the to be the basis. Oh be prime corresponds to the vector 0, 1 and 4 3rd. Okay, so now we have mapped each element of basis here. Indeed into the basis of the price. So now we can construct this meeting but the metrics will be just putting these com sectors us columns of the transformation matrix. The transformation matrix from b to b prime is defined, biden Matrix 0 -1. How 8 30 zero one and 4 30. So the transformation now we need to verify this. Indeed satisfy the formal of public transformation. Okay, we know that this kind of matrices should satisfy the following which we multiply the matrix to the victor in the basis of B. We should obtain this will be the same as applying here. The transformation of and then transform it into the base. Be proud. Okay, so let's see what happened with 11 times. So the left and right response to using the metrics that we zero mine how an eight third Euro one sports. We're going to be greed in a picture. The basis be. Well the this picture in B is taking also want time the vector you want plus also two times of extra Youtube engage. Did you don't remember A. one is a picture 1, 3 on YouTube is a back church -24. No here we need to take doctors in that basis. So basically we just need to multiply here But also one and out the result of this transformation. The matrix multiplication is the picture zero minus out spot one. How lost A. Two. I'm here one third. That's more replies to eight Out for one plus or all. I got to and is the result. Of course this is the basis because now what we need to do is take a better ex actually if you the base is this on the basis. B will be just for one and also to so we just need we need to pick this veteran here transform it to play the usual transformation and then transform into the basic deprived. So that is the right hand side. So first we need support we need to multiply by the vector. In this case is a picture will be also one -2 of the tooth and here will be three out for one Plus for Alpha two. When we apply this transformation to these vectors. Well we will obtain here be back to here, will be seven out to one bluff. six of the two. He will be -8 plus minus alpha one Plus two times out to. And the Okay, but this isn't the usual based right? We need to transform this into the basis prime. So when we transform this basically is to find the position. So we have this picture of the juan plus oh um minus over one or 2/2 0. We need to find the acquisitions here. Take me to one times you want plus beat it too. Okay and from this we obtained that the coefficient. I'm sorry this isn't the basis. The prime Can't be one plus B. two and last the three B. And from this we obtain of course what are going to be the corresponding coefficient in the basis? And we will obtain these basis. Okay, so basically that means that be the one will be zero, be that too will be minus Hope for one. Help look over to and beat three will will be equal two one third plane eight Of the 1-plus 4 times. And this corresponds to saying that this venture in the base. The prime is equal to the baxters zero minus of one house of the two. One third time. Eight of one plus 4 of that is the same as the left. Yeah

All right. So in this problem, work even the set of equations on That's ah, you want He was elected C one us for C two. And you, too. You close if I see one. Maya's three C two. Okay. And we need to transform these set of questions. Two, the form of A C wants you to and expressed by some leaner combination off B one and B two as the one plus you, too. All right, so in this problem, work even the set of equations, That's ah, you want He was elected C one us for C two, and you 21 I'll just rewrite this. Um, so Okay, one. Do you want us? Okay, too. You too. And 31 He won last year too. You, too. Okay to do that. Uh, first, I always question we have C one. You could see one from the right hand side to look inside. So that sexy one, Because, or c two minus b. You want Now we need to figure out a C two. So, by our second equation here, see, two will be five C one minus B two, divided by three so that it is we substitute C two. Bye, dizzy. But he's a fraction. We have five up. See one minus B two. You find it by three and minus B one. So this is same as 20. So we want our first question. We have C one. You could see one from right inside to look inside. So that's a C one because, or C two minus B you want Now we need to figure out a C two. So by our second equation here, seek to will be five c one minus B two divided by three. So that ISS my book sides by three at three greasy one and 20 c one and minus four. You too minus three you want and this is equivalent to So we could 20 way put 20 upsy one from my hindsight 11 site that we will get the negatives We substitute c two. Why is he needs a fraction? We have live up. See one minus two. You find it by three and minus B one So this is same as 20. So we won't apply both sides by three at three greasy one and 20 c one and minus four You too minus three do you want? And this is equivalent to So we could 20 way put 20 upsy one from my hindsight 11 site that we will get a negative 17 c one, which is same as for thinking for off B two minus three. So be one, and that's 17. C one equals four e too. Plus three off you won. So we keep going. Divide the both sides by 17 7 C one or over 17. You, too. Oh, just so I'll write down the wipers. So that's three over 17 of the wand, and we have for over 17 the Team C one, which is same. As for thinking for B two minus three, I'll be one and that's 17. C one equals four e, too, plus three of you won. So we keep going to buy the both sides by 17 off you, too. So that's for our C one. So we need to do the same for C two. C two is or I'll see to buy. Our first equation is equal to you. Want pussy? One. Now see one by our second equation is three up see, too plus you too. So that's C one it was or over 17. You too. Oh, just so I'll write down the wipers. So that's three over 17 of the one and we have for over 17. You find it by five. So that is be one plus C one. We substitute by this friction three years. Upsy too. Plus you too divided by five. And we first multiply five on both sides. We have 20 f c two off you too, so that's for our C one. So we need to do the same for C two. C two is or I'll see to buy. Our first equation is equal to be one pussy one. Now see one by our second equation is three up See, too Plus you too. It was five off the one last three. Upsy to class fee, too. That's 17 C two and five P one. Let's beat you on the right hand side and with divided hillside by 17. Yes, five over 17 B one plus won over 17 b two. You find it by five. So that is be one plus C one. We substitute by this friction three year autopsy too. Plus you too divided by five and we first Multiply five on both sides. We have 20 f C two. So how are matrix Will be? Well, first we have C one, which is three. We sorry. Three over 17 and or over. 17 54 17 and one over 17. It was five. B one lost three. Upsy to class for you too. That's 17 C two and five P one. Let's beat you on the right hand side and with divided hillside by 17. Yes. Live over 17. B one plus won over 17 B two, though. Here's our, um change of court in the matrix problem. B to C. No, part B. We need to find express kids. See Thio X because live off B one plus three a B two. So our matrix will be well, first, we have C one, which is three three. Sorry. Three over 17 and or over 17 5 17 and one over 17. So here's our, um change of court in the Matrix from B to C No part B. So the way we do that is too eyes to apply our me fix here. So Okay, so, uh, for X c, we first have Ah, He won undersea and we need to find express kids. See Thio X, because live off B one plus three a B two. So you, too in their seats and multiply Biomatrix. That is, um that is here. Our wishes here. That's five and three. Oh, the way we do that is too, uh is to apply our me fix here. So Okay, so, uh, for X C, we first have Ah, he won undersea. And Okay, so how do we find the one break it, Uh, the one Breck it see? So that is just from our matrix. So that is you want to see? Is so we have three over 17 and five over 70 three over 17 5 over 17. And for me, too, Bracket C would have you, too, in their seats and multiply Biomatrix. That is, that is here. Our question here. That's five and three. Okay, so how do we find the one break it, Uh, the one Breck it see over 17 and won over 17 and five and three. So we take out won over 17. So that's why 17 off matrix 3451 and vector. So that is just from our matrix. So that is you want to see is so We have three over 70 and five over 70 three over 17 5 over 17. And for me, too, bracket C, we have four over 53 So that's 1 17 while over 17 times. Victor 15 plus 12 27 and five times 5 25 plus 3 28 So this is our solution to the second part, which is ex record city.

Call them too. So we need to find Thea Person Need to find the change of courting the matrix from B to C and your work even be one is elective C one us or I'll see to it. Sorry, just or I'll see to. And you too is five C one minus three. Upset you. So by over your own given in the 15 given in the chapter 4.7 we I know that the Matrix changeup quote in the Matrix is you want record C and B two bracket See? And this is my ovary questions here. Active one and four and five and negative three. So this is our change of coordinates matrix. So the next part I need to find ex breakfast, see two giving question. There's X equals five, one less three. You too. So the way to do that is to notice, except C is Come, I don't need to see times. The vector here is a five three. So this will be our previous meetings. Negative one. Why? For negative Three times I three, which iss 10. He left


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