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.