Digital Image Processing

Assignment (1) Review

Due: Class time, Friday September 2

Note to students at the remote sites. If you have typed your answers, then e-mail me the electronic file as an attachment before class on due date (rt@cs.appstate.edu). Otherwise, submit the hardcopies to the local coordinator: Dr. Holliday at WCU, or Dr. Lea or Mr. Ridenhour at UNCG . Please write clearly.

 

Linear Algebra Problems

 

Consider the matrices A, B, C, and D below:

A=, B=, C=, and D=

 

1. Which matrices can be added to each other? If they may be added in more than one order, give both orders.

 

2. Which matrices can be subtracted from each other? If they may be subtracted in more than one order, give both orders.

 

3. Which matrices may be multiplied by each other? If they may be multiplied in more than one order, give both orders.

 

4. If matrices may be multiplied by each other in more than one order, does the 1st matrix x the 2nd matrix give the same answer as the 2nd matrix x the 1st matrix?

 

5. Find the matrix product C x D.

 

6. DT is given by . What is CT? (C is called a symmetric matrix.)

7. (-2) * DT is given by . What are 3 * C and (-4) * CT? Confirm your result with MATLAB.

 

Example: When you find the inverse of a matrix using the method outlined in class (see the review slides), the product of the diagonal elements of the original (left-side) matrix is the determinant of the matrix. To find det(C) and C-1, we use the following procedure:

Step1: Subtract 2 x row 1 from row 2, and 3 x row 1 from row 3. We get

Step 2: Subtract 5 x row 2 from row 3. We get

 

Step 3: At this point, we know the determinant: 1 x (-1) x 18 = -18. To find the inverse, we must turn the left side matrix into the identity. We start by multiplying row 2 by -1 and dividing row 3 by 18. We get

 

Step 4: Now we subtract 5 x row 3 from row 2, and 3 x row 3 from row 1, getting

 

Step 5: To complete the process, we subtract2 x row 2 from row 1, getting

. Thus C-1 = 1/18 ,

 

which you can check by multiplying C by C-1, and seeing whether you get the identity matrix. Do this multiplication in MATLAB.

 

8. Find the determinant and inverse for each of the following matrices using the method described above.

A= B= C=, and D=

 

 

9. Consider the vectors u, v, and w below.

u = , v = , and w =

 

The DOT product of u and v is 1, i.e.; . What are

 

Is

 

10. u x v = -5i + 5j k. What are the CROSS Products?

v x u?

v x w?

u x w?

is u x v = v x u?

 

 

 

 

Statistics and Probability Problems

11. X = {2, 2, 3, 4, 5, 5, 6, 7, 8} and Y = (-3, -1, 1, 3, 5, 7, 9, 11}. The mean of X is 4.6667, the median of X is 5, the range of X is 6, the mean deviation of X is 1.7, the sample standard deviation of X is 2.12, and the variance of X is 4.5.

 

What are the corresponding quantities for Y?

 

12. The probability of each of the values in X is given by 2 or 5, 2/9 each; 3, 4, 6, 7, or 8, 1/9 each. The probability distribution function (starting at 2) has the values 2/9, 1/9, 1/9, 2/9, 1/9, 1/9, 1/9. The cumulative distribution function has the values (starting at 2) of 2/9, 3/9, 4/9, 6/9, 7/9, 8/9, 9/9.

 

What are the corresponding values for Y?