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Let $\mathbf{v}_{1}=\left[\begin{array}{l}{1} \\ {0}\end{array}\right], \mathbf{v}_{2}=\left[\begin{array}{l}{1} \\ {2}\end{array}\right], \mathbf{v}_{3}=\left[\begin{array}{l}{4} \\ {2}\end{array}\right], \mathbf{v}_{4}=\left[\begin{array}{l}{4} \\ {0}\end{array}\right],$ and $\mathbf{p}=\left[\begin{array}{l}{2} \\ {1}\end{array}\right] .$ Confirm that$\mathbf{p}=\frac{1}{3} \mathbf{v}_{1}+\frac{1}{3} \mathbf{v}_{2}+\frac{1}{6} \mathbf{v}_{3}+\frac{1}{6} \mathbf{v}_{4}$ and $\mathbf{v}_{1}-\mathbf{v}_{2}+\mathbf{v}_{3}-\mathbf{v}_{4}=\mathbf{0}$Use the procedure in the proof of Caratheodory's Theorem to express $\mathbf{p}$ as a convex combination of three of the $\mathbf{v}_{i}$ 's. Do this in two ways.

$p = \frac { 1 } { 2 } v _ { 1 } + \frac { 1 } { 6 } v _ { 2 } + \frac { 1 } { 3 } v _ { 3 }$

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 3

Convex Combinations

Vectors

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we can check that. This is expression is correct. Just by doing the corresponding products on the some eso Here we have a p written as a linear combination off the B one b two B three, before on the coefficients of these linear combinations are going to be called. She wanted to see three and C four on. What we want to do is to write p as a linear combination, but involving less factors. So for that, we're going to follow Kathy other East procedure so you can go check the details in the chapter on were given this expression to Sophie one minus V. Two plus feet, three minus before sirrah. However, we're going to multiply by minus one. So we have minus 131 plus B two minus 53 plus before equals to Zira. I wrote it this way because I want to. Well, I will make reference to this coefficients on. I'm going to call the thing one thing to think. Three on 34 Okay. So e multiplied by minus one. Because I wanted to have the four have positive coefficient for the more I wanted see four over the four, which is in this case 1/6 to be less than or equal tendency to over the two, which is one third. I was interested in C two over the two because the two is also positive. So now I'm going to define this numbers they want or B I equals C I minus C 4/84 times three I. So we're going to compute them. The one is gonna be a C one, which is one third minus c for a birthday four, which was 1/6 time. Stay one which is minus one B two city, which waas one third, minus 16 times the two which Waas one be tree is going to be C three in this case 1/6 minus 16 times in three, which is minus one on the same for before before is going to be 1/6 minus 16 10 54 which is one on these numbers are one half 16 one third and Syria On. With this numbers, we can write p as a near linear combination. The in this case, it's gonna be one half off. Be one pause 1/6 off feet, too, plus one third off feet three. No, I'm going to write the fees. Explicitly. This is one half of the victors. She 10 +16 off Vector 12 on plus one. Third time's a factor for two on board. Yes. So we have done half of the problem. We need to find another way to express this vector p. Again, We're going to do it by using this method with it before. So for our second expression, we're going to let the big one and feet too, as they were before. But we're going to interchange feed three on before, So V three is going to be 40 on before is gonna be the vector 12 Now you can verify that. Be one minus feet too. Minus 33 plus before equals to zero eso this expression is gonna give us the new the eyes on notice that since the coefficient C three and C four, we're both 1/6 then the sea ice. They're going to be the same as before. Um, so we like this 31 minus 50 minus 53 plus fee for equal zero Because this way, see forever. The four is again. 1/6 on this is less than or equal than see one over the one which is one third we care about you whenever they want. Because they want was Thea other positive coefficient in this expression. So now we have everything to compute the B eyes again. B one is one third, minus 16 times the B one which in this case is one B two is gonna be one third, which is it to minus 1/6 times did, too. Now the two is minus one Be tree is going to pay one sexed minus 16 times the three we choose minus one on before is gonna be 1/6 minus 16 times before, which is one now. These numbers are before is going to be zero B three is one third B one is when sixth on B two is one half. So our second expression is P equals to 1/6 off 31 plus one half off feet too. Plus when third off feed three. But now here you gotta be careful, because remember, we interchange the roles of the three and before so writing the vectors explicitly, we have p equals 16 of the victor 10 plus one half of the victor 12 plus one third of the victor Forests. Hero on this is another way to write P s a linear combination off they for or, you know, victories.

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