5. The following series of questions explores the relationship between the standard dot product on R" (geometry) and linear operators on R" (linear algebra). Let U be a n x n matrix, and think of it as a linear operator on R" in the usual way (via matrix mutliplication). A n x n matrix is called orthogonal if its columns form an orthonormal set. (a) Show that U = cos e - sin d sin A cos # is an orthogonal matrix, where 0 is any number. (b) Show that U is an orthogonal matrix if and only if U"U = I [HINT: use the definition of matrix multiplication.] (c) Show that if U is an orthogonal matrix, then for any vectors x, y, Ux . Uy = X . y HINT: Think of the dot product as matrix multiplication and use the orthogo- nality of U.] (d) Show that if U is an orthogonal matrix, then for any vector x, [HINT: Use the definition of the length of a vector and part (c).] (e) Show that if U is an orthogonal matrix, then for any vectors x, y, the angle Z(x, y) between x and y is equal to the angle Z(Ux, Uy) between Ux and Uy. [Hint: Use the definition of angle and parts (c) and (d).] (f) Show that if U is an orthogonal matrix, then for any vectors x, y, dist(x, y) = dist (Ux, Uy). [Hint: Use the definition of distance and parts (c).] (g) Use the fact that orthogonal matrices must preserve lengths and angles to find a 2 x 2 matrix A that is not orthogonal. Verify that it is not orthogonal. In summary, linear operators defined by orthogonal matrices "preserve" the dot prod- uct, and hence all geometric notions. In a certain sense, they are the linear operators that are compatible with the geometry.The National Center for Health Statistics published data on heights and weights. We obtained the following data from 10 randomly selected males 8 - 12 years of age. ght 69 71 70 67 75 66 73 71 68 65 ight 151 159 160 153 198 126 174 185 143 175 a. If the researcher uses "Height" to predict "Weight" then the response variable is (2pts] b. Use your calculator to create a scatter plot. Copy the scatter plot below (draw it). Make sure to label the horizontal axis and vertical axis, and provide the minimum and maximum values (displayed on your graph on your calculator) on each axis. [4pts] c Is the association between the two variables positive or negative, or no association? (circle one) [2pts] d. Use your calculator to find the regression line (or line of best fit) and the correlation r. (Make sure something has a hat on it.) Regression Equation: [4 pts] Correlation r= [2 pts] e. Use the equation in d. to find the predicted weight for a man who is 69" tall. Show work and box final answer. [3pts] F. Compute the residual associated with the man who is 69" tall, Show work and box final answer. [2pts] g. Identify the slope of the regression line: Interpret it using the context of the problem. (Write a sentence and mention both men's heights and weights.) (3pts]a) Use this information to construct a payoff matrix for this game. Use the following format. I've filled in one of the hard boxes to get you started: B: High B: Low A: High A: Low ($2m, $6m) b) What is firm A's best response if firm B produces High? What is A's best response if firm B produces Low? c) Use Nash equilibrium to explain why both firms end up producing the High level of output, even though they can earn higher profits if both produce low. d) Does either firm have a dominant strategy? Explain. 4) Now, we're going to look at a Stackelberg equilibrium in this industry. The cost and demand functions are all exactly the same as in #2, but now firm A gets to set its output level first, and then firm B chooses output after observing firm A's output. a) Firm B still has the same best response function from 2a/2b. Firm A knows firm B will follow this best response function in response to A's output. Use this to calculate firm A's profit maximizing output decision. Remember how the total revenue function you calculated in 2A had On in it? Now you can use B's best response function to replace this Os with a function of QA. This will give you A's total revenue as just a function of Q, From here, just use MR. = MC, as normal. b) Based on your answer to 4a, calculate B's profit maximizing output (B's best response to QAfrom fa), the price of the good, and the profits each firm makes. 5) In this question, we're going to use a game tree and backward induction to analyze a Stackelberg problem. In this case, each firm has 3 options: to produce a low level of output, a medium level of output, or a high level of output. The payoff's to each firm given each firm's output choice are: B: High B:Medium B: Low A: High (0,0) (75.50) (1 12.50, 56.25) A: Medium (50.75) (100. 100) (125, 93.75) A: Low (56.25, 112.50) (93.75, 125) (1 12.50, 1 12.50) a) What is the Nash equilibrium when both firms choose their output at the same time? Explain. You must prove that your answer satisfies the definition of Nash equilibrium