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 — gibson:teaching:fall-2014:math445:hw3solns [2014/10/07 10:43] (current)gibson created 2014/10/07 10:43 gibson created 2014/10/07 10:43 gibson created Line 1: Line 1: + ====== Math 445 HW3 solutions ====== + <​code>​ + John Gibson + Math 445 + HW3 solutions + Oct 7, 2014 + + + Problem 1: Given the vectors x=[3 7 2 9 0] and y=[7 10 2 8 13], what + would be the Matlab output for the following expressions?​ Think + through what the answer should be, write it down, and then try it out + in Matlab. If you got anything wrong, figure out what your mistake was + and why Matlab gave the answer it did. + + (a) x > y + [0 0 0 1 0] + + (b) y < x + [0 0 0 1 0]    ​ + + (c) x == y + [0 0 1 0 0] + + (d) x <= y + [1 1 1 0 1] + + (e) y >= x + [1 1 1 0 1] + + (f) x & y + [1 1 1 1 0] + + (g) x & (~y) + [0 0 0 0 0] + + (h) (~x) & (y) + [0 0 0 0 1] + + (i) x | y + [1 1 1 1 1] + + (j) xor(x,y) + [0 0 0 0 1] + + (k) (x > y) & (y < x) + [0 0 0 1 0] + + Problem 2: Write down Matlab expressions for the following. You can + assume that a,b,c are logical variables, x,y,z are double-precision + numbers, u,v,w are vectors of the same dimension, and A,B,C are + matrices of compatible sizes. + + (a) Both a and b are true. + a && b + + (b) Neither a nor b is true. + ~a && ~b + + (c) Either a and b are both true, or b and c are both false. + (a && b) || (~b && ~c) + + (d) Either x equals y, or x is not equal to z. + (x == y) || (x ~= z) + + (e) x, y, and z are all equal. + (x == y) && ​ (x == z) && (y == z) + + (f) None of the components of u equal the corresponding components of v. + ~any(u == v) + + (g) Each component of u is the same as either the same component of v or w. + all(u == v) + + (h) The vector whose components are the polynomial 3u^2 - 5u + 6 evaluated ​ + at each of the components of u. + 3*u.^2 - 5*u + 6 + + (i) The matrix product AB. + A*B + + (j) The matrix whose elements are the product of the elements of A and B. + A.*B + + Problem 3: A theater has a seating capacity of 900 and charges \$2.50 + for children, \$4 for students, and \$5.50 for adults. At a certain + screening with full attendance, there were half as many adults as + children and students combined. The total money brought in was + \$3825. How many children, students, and adults attended the show? + show? + + The equations are + + c +  s +    a = 900 + 2.5c + 4s + 5.5a = 3825 + 0.5(c + s) = a + + or + + c +    s +    a = 900 + 2.5c +   4s + 5.5a = 3825 + 0.5c + 0.5s -    a  = 0 + + To put this in matrix-vector notation, let the column vector x + have components [c  s  a]'. Then + + A = [1 1 1; 2.5 4 5.5; 0.5 0.5 -1]; + b = [900 3825 0]'; + + x = A\b + + x = + 150 + 450 + 300 + + So 150 children, 450 students, and 300 adults attended the show. + 