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Problem #1: On a certain island, there is a population of snakes, foxes, hawks and mice. Their populations at time / are given by s(t), f(t), h(t), and m(t) respectively. The populations grow at rates given by the differential equations s's-1-3- = ·m fsf-m + h = +" m² = $s-98f-2h-1 m Putting the four populations into a vector y(t) = [s(t) f(t) h(t) m(t)], this system can be written as y'= Ay. Find the eigenvectors and eigenvalues of A. Label the eigenvectors x₁ through x4 in order from largest eigenvalue to smallest (the smallest being negative). Scale each eigenvector so that its first component is 1. When you have done so, identify the eigenvector whose fourth component is the largest. What is that largest fourth component? Problem #1:
Problem #2: Continuing from the system of differential equations from Problem 1, each eigenvector represents a grouping of animals that changes with simple exponential growth or decay. The exponential rate of growth or decay is given by the corresponding eigenvalue. Because the matrix d is invertible and diagonalizable, any initial values for the animal population can be written as a combination of these four special groupings that each grow exponentially by their eigenvalue. Consider the initial population y(0) = [7 3 2 29]. Solve for constants c1 through c4 in order to write y(0) c1 x1 + 2x2 + C3 X3 + C4X4 where x1 through x4 are the eigenvectors as detailed in Problem 1 (i.e., the eigenvectors in order, and scaled so that the first component is 1). Enter the values of c1, c2, c3, and c4, separated with commas. Just Save Submit Problem #2 for Grading Problem #2 Attempt #1 Attempt #2 Attempt #3 Your Answer: Your Mark: Problem #2:
Problem #3: Based on the system of differential equations from Problem 1, with the initial population from problem 2. find the function for the population of mice, m(t). Problem #3: Enter your answer as a symbolic function of t, as in these examples Just Save Problem #3 Attempt #1 Attempt #3 Your Answer: Your Mark: Submit Problem #3 for Grading Attempt #2
Problem #4: A matrix with complex entires is called aitary if A=4", where A4" is the conjugate transpose described in the Tutorial 4 file. Which of the following matrices are unitary? *#[HA] @[1] @[H D I-i 1+i 1-1- Note: Testing matrices for equality is always subject to the usual innacuracies in floating point arithmetic. So for the purpose of this problem, you can consider two matrices to be equal if their entries agree to at least 4 decimal places. Warning! Don't forget the constants in front of each matrix. They are crucial for this problem (A) (i) only (B) (u) only (C) none of them (D)) and (m) only (E) (iii) only (F) (i) and (ii) only (G) (ii) and (iii) only (H) all of them Just Save Submit Problem #4 for Grading Problem #4 Attemot #1 Attemot #2 Attempt #3 Your Answer: Your Mark Problem #4: Select
Problem #5: (a) Let u-(-2,-9, -6, -6) and v-(3, -10, 6, 5). Findu - proj,u Problem #5(a): Problem #5(b): Note: You can partially check your work by first calculating proj,u, and then verifying that the vectors proj,u and u-proj, u are orthogonal. (b) Consider the following vectors u, v, w, and z (which you can copy and paste directly into Matlab). u= [-3.2 6.3 -3.6), v= [5.3 -8.4 5.4], w = [-2 8.7 -9.11, z = [-9.6 4.6 1.31 Find the determinant of the following matrix. uw u z To check your work, you can also calculate the following quantity: (uxv) (wxz). This quantity, and the determinant of the above matrix, must always be equal. Just Save Submit Problem #5 for Grading Problem #5 Attempt #1 Attempt #2 Your Answer: 5(a) 5(b) Your Mark: 5(a) 5(b) 5(a) 5(b) 5(a) 5(b) Attempt #3 5(a) 5(b) 5(a) 5(b)