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For $H_{2}, \gamma=\frac{7}{5}$ $p_{1} V_{1}=p_{2} V_{2}, p_{3} V_{3}=p_{4} V_{4}$ $p_{2} V_{2}^{\gamma}=p_{3} V_{3}^{\gamma}, p_{1} V_{1}^{\gamma}=p_{4} V_{4}^{\gamma}$ Define $n$ by $V_{3}=n V_{2}$ Then $p_{3}=p_{2} n^{-\gamma}$ so $p_{4} V_{4}=p_{3} V_{3}=p_{2} V_{2} n^{1-\gamma}=p_{1} V_{1} n^{1-\gamma}$ $p_{4} V_{4}^{\gamma}=p_{1} V_{1}^{\gamma}$ so $V_{4}^{1-\gamma}=V_{1}^{1-\gamma} n^{1-\gamma}$ or $V_{4}=n V_{1}$ Also $\quad Q_{1}=p_{2} V_{2} \ln \frac{V_{2}}{V_{1}}, Q_{2}^{\prime}=p_{3} V_{3} \ln \frac{V_{3}}{V_{4}} n^{1-\gamma}=p_{2} v_{2} \ln \frac{V_{3}}{v_{4}}$ Finally $\eta=1-\frac{Q_{2}}{Q_{1}}=1-n^{1-y}=0.242$ (b) Define $n$ by $p_{3}=\frac{p_{2}}{n}$ $p_{2} V_{2}^{y}=\frac{p_{2}}{n} V_{3}^{\gamma}$ or $V_{3}=n^{1 / \gamma} V_{2}$ So we get the formulae here by $n \rightarrow n^{1 / \gamma}$ in the previous case. $\eta=1-n^{(1 / \gamma)-1}=1-n^{-\frac{2}{7}} \sim 0 \cdot 18$

In this recipe. Rector of Elephants and the office is recorded s square ICAP. Let's do this, Jacob. That's nine. The sliver minus to kick up than we have. Let's see off off a sequel. Toe three. And and she does so Fouras ago. Madness Name freedom No. Went to 100 d Barias off the off the office. This is a girdle. Using changing leaders off she office with your brother was you'd ask of us. This is a good all the way. D s off the V profess there is this ecology office in order to permit she does off this. Where s it going? Four. This is a quote Ruedas Rodeo. The universe is de vie. Do you buy the year? So this is square like this. Two s, Jacob minus nine. I said I minus two cake. This is going to see off us Different Reggie Judaism as Judah's off. This is a proto toe s I kept. That's to Jacob. This 18 steady power minus three K cup is visible. CEO who and what she does off Wolf. This is according Toa Geo whore. Jacob has to Jacob. That's 18 in Tokyo. Who are the problem is three cuc up. Pretty fair visitors of Forest Man last night. This is a Karoo NGO or s three 203 I cab was to Jacob. This 18 and to three by minus three. Minus name. So this is a photo six I cab the spool. Jacob. Yes, 18. That's to write three K cab into minus nine. This is a code of minus 54 a cab when the Sultan Jacob on a six Caykur. Okay, we have a small can. Extending it instead off minus nine. We have plus nine years a here to B minus, so minus minus off. Minister Li plus six k up. So this is the value photo, everybody in terms of it.

In each parts were asked to find the coast side at an angle of data. Now in part a data is the angle between the victory you with components 13 negative 54 And the Is the vector 2 -341 just in our four. Well, to find the angle co sign of data, let's first find the inner product of U. And V. This is the sum of the product of corresponding components which is 2 -9 -20 Plus four, Which is -23. Right. We also want to find the norms of U. And V. So let's find the normal view squared. This is the sum of the squares of the components of you which is one plus nine plus 25 plus 16 Which is 51. And let's find the normal v squares again some of the squares of the components of the which is four plus nine plus 16 plus one which is 30. And therefore it follows that the co sign of her angle data is equal to thanks. Right? So our inner products negative 23 Over the normal view which is the square root of 51 times in R. V. Which is the square root of 30. We want to simplify and we get negative 23 over three Times The Square Root of 170. That's the coastline of our angle. Theta. Then in part B. It seems so much. Yeah. Data is the angle between the matrices. A. With entries 987654 And be with entries 1 234 56 Things where the inner product of A. And B. These two matrices is defined to be the trace of the transposed times A. So assuming that this is in fact a valid in our product. How are you going to find a coastline of this angle? More than well, first of all notice and the rewrite our inner product using algebra. So like the trace of being transposed A. So I like yeah well this is going to be the sum from I equals one. Crazy. You know, she's it's like in this case m. Is it's like and she up to six. This uh it's just A. I. J. Sorry? Nice dude. He's going yeah that's Mhm. Yeah. Saca's. So if the sum from I equals not six. Uh huh. 12 two. Yeah. Some from J equals 123 of A. I. J. Times B. I. J. Six bus. So this is equal to one time is nine which is nine Plus eight times 2 16 And so on. So we get 21 plus 24 plus 25 Plus 24 again. And this is 119. So this formula really does come in handy for this enterprise. Some trees Now you want to find the norms of A and B to do this. We'll find the norm squared of matrices A and B. Interest in the norm, swear today is by definition being a product of A. Which itself? Which? All right, bo this is the same as the some from I equals 1 to 2 of the some from J equals 123 of a I J squared. And therefore this is the sum of the squares of all the elements of A. Yeah, I'm going to get. Mhm. So nor they squared is equal to nine squared plus eight squared plus seven squared plus six squared plus five squared plus once yes or squared, Which simplifies through 271 And therefore the norm of a is the square root of 271. Likewise, the number of B squared is also the sum of the squares of the entries of the so it's the enterprise to be with itself which is one squared plus two squared plus three squared plus four squared plus five squared plus six squared, Which simplifies to 91. And therefore the normal B is the square root of 91, you see. Yes. Therefore the co sign of our angle data. If we use the formula cosign. Theta is inter product of you? Ve over the product of the magnitudes of directors. Well, this is going to be the enterprise with A. & B., which was 119 over the square root of 271 Times The Square Root of 91. Yeah, sure. Hey, yes, Yeah.

We are given expressions the rest. To expand these expressions. In part they were given The inner product of five. You one plus 8 U. two and 61 -72 by linearity. Oh we can write this as five. U. One 61 Plus eight. U. 2 six. V. One Plus five. U. 1 negative 72 Plus eight. U. 2 negative 72. And then using modernity. This is 30 times In a product of you warning V one minus 35 times in a product to you. One with the two Plus 48 times in a product You to be one you should plus or minus 56 times the inner product you too. D. Two. Right? In part B. We are given the inner product Of three. U. Plus five E. And for you -60. Once again I'll use linear very and homogeneity. So this becomes three times 4 or 12. You in a product with you -3 times six or 18. Enter product of U. And V Plus I can score or 20 inner product to be with you -5, 10, 6 or 30. And our product view is V. And now using symmetry Inter product of U. V is same as inner part of the with you. So this is the same as 12. enter product you with you Uh plus two times in our product you with me minus 30. Inner product B. With B. Then in part C forgiven The norm of two U -3 D squared First I'll rewrite this using our products. This is the inner product of two U -3 d. with two U -3 d. Using linearity. This is the same as two times two or 4. Inner product to you. If you -2 times three or 6. Inner product uh You with me minus six. Enter Product v. With you plus nine. Inner product V with V. And then, by symmetry, this simplifies to four inner product to you. With you -12. Inner Product, you would be Plus nine in your product is in you with fees, which of course we could also, Right As four times the norm of you square -12 times the inner product you with V Plus nine times the normal v. Swear. Mhm, mm hmm. Yeah, what just

Restaurants that are three vectors you with coordinates 1-4 V. with coordinates to -35 And W at coordinates 4 to -3. In our three. In part they were asked to find the dot product of the U. And V. To do this. We simply multiply corresponding components and add so that you don't give me Is to -6 course 20 which is 16. Yes. In part B we're asked to find you done W. Once again multiply corresponding components and add we get and two. No, it's not cute. Four plus four -12 which is -4. And park. See we're asked to find a V. W. W. So once again multiply corresponding components and add we have eight minus six minus 15. Uh huh. And negative 13. Just just coffee on. In part B. We're asked to find you plus B dotted with W. Genesis. Okay. First we'll find you plus V. Children by adding components. So we have one plus two. Is three. You close to be is 3 -19? Yes. Donald W. And at this point will multiply corresponding components and add Yeah. So this gives us 12 -2 -27. Yes. Which is maybe just 17? Yes. Alternatively, you get the same answer. You plus the data. W by the distributive itty of the dot product is you Don T v f W plus V W W which we already saw was negative for And then -13, which is also -17. Just another method then in part whereas to find the norm of you single cells, I just first we'll find the normal view squared by squaring the components of you and adding. So we get one squared plus two squared plus four square, which is one plus four plus 16 Which is 21 and therefore mr the norm of you Is the Square Root of 21. Yes. Then in part F for us to find the enormous V. Similarly, let's find the number of B squared first by squaring components as we and adding this is going to be two squared or four plus three squared, or nine Plus five squared, or 25, which is 38 and therefore it follows that the normal V is the square root of 38. It's


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