MA-305 Homework 6

Due at 4:05pm, Tuesday, November 18, 1997

Solution

You may do the calculations necessary for these problems either by hand or with Maple. Please submit a handwritten solution, or email an ASCII/Postscript/html document to the TA. (You may also submit through email which attached your Maple worksheet.)

1. *Let W = span { ( 1, 2, -1), ( -1, 3, 2)}.
   (a) Find a basis for the orthogonal complement of W.
   (b) Describe the orthogonal complement of W geometrically. (You may use a verbal or pictorial description.)

   Solution:

   (a) { ( -7, 1, -5)}.
   (b) A line orthogonal to W in a three dimentional space.

2. For the given matrix, find a basis for its row space, find a basis for its column space, and determine its rank.

   (a)	[	2	4	1	2	1	]
   	[	4	2	3	2	2	]
   	[	2	-2	2	0	1	]
   	[	6	6	4	4	3	]

   
   

   (b)	[	1	-2	4	1	]
   	[	3	1	-3	-1	]
   	[	5	-3	5	1	]

   Solution:

   (a) {( 1, 0, -1, 1), ( 0, 1, 1, 1)}, rank = 2.
   (b) {( 1, 0, 2), ( 0, 1, 1)}, rank = 2.

3. Let u = ( 1, 2, 3) and v = ( -1, -1, 2). Compute:
   (a) u·v, the scalar product of u and v.
   (b) || u - v ||, the Euclidean norm of u - v.
   (c) the angle between u and v.
   (d) the projection of u onto v.

   Solution:

   (a) 3.
   (b) 3.741657387.
   (c) 1.237323155 radians or 70.89°.
   (d) ( -1/2, -1/2, 1).

4. Consider the following test runs on merge sort:

   math% testmsort 15 30 101
   Input array
   3 24 15 2 21 29 8 17 29 2 27 1 17 20 27 
   Indirect Mergesorted array after 44 comparisons
   1 2 2 3 8 15 17 17 20 21 24 27 27 29 29 
   math% testmsort 20 40 101
   Input array
   23 14 25 2 11 39 18 7 29 12 17 21 7 20 37 11 21 
   36 6 8 
   Indirect Mergesorted array after 62 comparisons
   2 6 7 7 8 11 11 12 14 17 18 20 21 21 23 25 29 
   36 37 39 
   math% testmsort 30 60 77
   Input array
   11 54 5 2 59 11 34 3 5 12 5 53 3 40 17 15 37 56
   18 12 10 41 38 36 23 49 43 5 42 59 
   Indirect Mergesorted array after 115 comparisons
   2 3 3 5 5 5 5 10 11 11 12 12 15 17 18 23 34 36 
   37 38 40 41 42 43 49 53 54 56 59 59 
   math% testmsort 50 60 79
   Input array
   16 36 0 32 17 38 0 57 24 43 44 42 30 42 52 37 54
   45 57 49 7 37 20 40 45 21 20 12 52 29 47 8 5 48 
   33 15 26 33 12 42 16 48 16 38 30 8 16 25 53 5 
   Indirect Mergesorted array after 224 comparisons
   0 0 5 5 7 8 8 12 12 15 16 16 16 16 17 20 20 21 
   24 25 26 29 30 30 32 33 33 36 37 37 38 38 40 42 
   42 42 43 44 45 45 47 48 48 49 52 52 53 54 57 57 
   

   We know that the number of comparisions of merge sort is the model:
   C(n) = c₀ + c₁ n + c₂ n ln n

   Using least square fits, determine the constants c₀, c₁, and c₂ and then estimate the number of comparisions taken for n = 10,000.

   Solution:

   C(n) = 1.387891674 x ln( x ) - 0.9555230720 x + 0.5661760422.
   C( 10,000 ) = 118274.8827.