Problem Set 6, Due: Thursday, Oct. 30, 2008
(late papers OK until 1:00 Friday)

Some Graphics. Our lecture notes on Vectors and Least Squares may be useful here.
1. a). For which values of the constants a and b are the vectors U = (1+a, -2b, 4) and V = (2, 1, -1) perpendicular?
   b). For which values of the constants a and b is the above vector U, perpendicular to both V and the vector W = (1, 1, 0)?

2. Let X = (3, 4, 0) and Y = (1,-2,1).
   a). Write the vector Y in the form Y = cX + V, where V is orthogonal to X. Thus, you need to find the constant c and the vector V.
   b). Compute ||X||, ||Y||, and ||V|| and verify the Pythagorean relation   ||Y||² = ||cX||² + ||V||².

3. Linear maps F(X) = AX, where A is a matrix, have the property that F(0) = A0 = 0, so they necessarily leave the origin fixed. It is simple to extend this to include a translation,
   F(X) = V + AX,
   where V is a vector. Note that F(0) = V.

   Find the vector V and the matrix A that describe each of the following mappings [here the light blue F is mapped to the dark red F].
   a). b).

   c). d).

4. a). Find a 2 × 2 matrix that rotates the plane by +45 degrees (+45 degrees means 45 degrees counterclockwise).
   b). Find a 2 × 2 matrix that rotates the plane by +45 degrees followed by a reflection across the horizontal axis.
   c). Find a 2 × 2 matrix that reflects across the horizontal axis followed by a rotation the plane by +45 degrees.
   d). Find the inverse of each of these maps.

5. a). Find a matrix that rotates the plane through +60 degrees , keeping the origin fixed.
   b). Find a 3 × 3 matrix that rotates R³ as follows: it keeps the x₁ axis fixed but rotates the x₂ x₃ plane by 60 degrees.

6. A square matrix P with the property that P² = P is called a projection. The point of this problem is to show that the name is justified, at least in the special case of a 2 × 2 matrix P that is neither the zero nor identity matrix.
   The simplest example of a projection is the orthogonal projection of the plane onto the horizontal axis, so P: (x₁, x₂) --> (x₁, 0). Its matrix form is (clearly!)

                      |  1   0 |
                  P = |        |
                      |  0   0 |

   
   a). A slightly less obvious example to think of is the matrix Q defined by

                    |  1/2  -1/2 |
               Q =  |            |
                    | -1/2   1/2 |

   i). Verify that Q² = Q.
   ii). Find (non-zero) vectors U and V with the property that QU = 0 and QV = V.
   iii). Write W = (1,2) in the form W = rU + sV (so find r and s). [In the language of linear algebra, we are using U and V as new coordinates for the plane.] Show that QW = sV, so Q kills the U component of W but leaves the V component unchanged. Drawing some sketches may help clarify this.
   b). More generally, for any 2 × 2 projection P (other than the two trivial cases P = I and P = 0), show there are non-zero vectors U and V with the properties that PU = 0 and PV = V; thus P kills U and leaves V unchanged. Also, show that one can write any vector X as X = W + Z, where PW = W and PZ = 0 (of course W and Z both depend on X).

   Evaluating Data

   The next two problems show that care must be taken when making statements about ratios and probabilities. Simpson's paradox. supplies an example that arises frequently: If A performs better than B in every category, must the overall performance of A necesarily be better than B?
   These examples are from D'Angelo and West Mathematical Thinking, Second ed, Chapter 9, Prentice-Hall, 2000
   You might find the example Sex bias in grad school admissions? [click here] useful. In particular, please compute the weighted averages as was done in that example.

7. Discuss the following table. Which airline has a better "on time" record?

   	Alaska Airline		America West
   Destination	% on time	# arrivals	% on time	# arrivals
   Los Angeles	88.9	559	85.6	811
   Phoenix	94.8	233	92.1	5255
   San Diego	91.4	232	85.5	448
   San Francisco	83.1	605	71.3	449
   Seattle	85.8	2146	76.7	262
   Total	86.7	3775	89.1	7225

8. Spoiled Fruit. Which of the two farms does a better job at minimizing the amounr of spoiled fruit? Justify your response.

   	Eva's Farm			Tom's Farm
   	total	spoiled	% spoiled	total	spoiled	% spoiled
   melons	1,000	100	10%	5,000	70	1.4%
   oranges	150,000	1,000	0.67%	10,000	50	0.5%
   Summary	151,000	1,100	0.73%	15,000	120	0.8%