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MATLAB Math Assignment #1
1 Revised 1/26/05
LAB 1: Matrix and Vector Computations in MATLAB
In this lab you will use MATLAB to study the following topics: How to create matrices and vectors in MATLAB. The commands to do this are short and easy to remember because MATLAB specializes in matrix computations and uses standard linear algebra notation. How to manipulate matrices in MATLAB and create matrices of special types. How to add matrices and multiply a vector by a matrix by MATLAB. How to nd the reduced row echelon form of a matrix by MATLAB.
Preliminaries
Reading from Textbook: Before beginning the Lab, read through Sections 1.1, 1.2, 1.3 and 1.4 of the textbook and work the assigned homework exercises for these sections. Demonstration of MATLAB: The basic mode of MATLAB is interactive. After you start the MATLAB program and obtain the prompt >>, you type commands that MATLAB then executes when you press the Enter key. If you have never used MATLAB before, we suggest you type demo at the MATLAB prompt now. Click on Desktop Environment and run the playback les (you may want to return to this demo later as you work through the lab and prepare your writeup). Then move on to the demo of Matrices. and Run Basic matrix operations. This gives a slide show that demonstrates how matrices are entered and displayed. Also run the demonstration Matrix manipulation. Diary File: Now that you have seen a demonstration of MATLAB, you can begin to work through this assignment. You will need to record the results of your MATLAB session to generate your lab report. Create a directory (folder) on your computer disk to save your MATLAB work in. Then use the Current Directory eld in the desktop toolbar to change the directory to this folder. Now type diary lab1.txt followed by the Enter key. Now each computation you make in MATLAB will be saved in your directory in a text le named lab1.txt. You can then edit this le using your favorite text editor. When you have nished your MATLAB session you can turn o the recording by typing diary off at the MATLAB prompt (dont do this now). If you want to stop your MATLAB session before completing a lab assignment, you can reopen the diary le the next time you start MATLAB. If you use the same le name, the results of your new MATLAB session will be written at the end of the old diary le. You may prefer to use dierent names (such as lab1a.txt, lab1b.txt ) for each session on an assignment, and then merge the les when you prepare your lab write-up with a text editor. Of course, for the other labs you will change the lename to lab2.txt, and so forth. Lab Write-up: Now that your diary le is open, type the comment line % Math 250 MATLAB Lab Assignment #1 at the MATLAB prompt. Type format compact so that your diary le will not have unnecessary spaces. Put labels to mark the beginning of your work on each part of each question, so that your edited lab write-up has the format % Question 1 (a) . . . . . . % Question 1 (b) . . .
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and so on. Be sure to answer all the questions in the lab assignment. Insert comments in your diary le as you work through the assignment. Random Seed: When you start your MATLAB session, initialize the random number generator by typing rand(seed, abcd) where abcd are the last four digits of your Student ID number. This will ensure that you generate your own particular random vectors and matrices. BE SURE TO INCLUDE THIS LINE IN YOUR LAB WRITE-UP The lab report that you hand in must be your own work. The following problems use randomly generated matrices and vectors, so the matrices and vectors in your lab report will not be the same as those of other students doing the lab. Sharing of lab report les is not allowed in this course.
Question 1. Creating Matrices and Vectors
(a) The most direct way to create a matrix in MATLAB is to type the entries in the matrix between square brackets, one row at a time. To separate the entries in the same row, type a comma or press the space bar. To indicate the beginning of a new row, type a semicolon or press the Enter key. Try this by typing A = [1 2; 3 4; 5 6] (followed by Enter). MATLAB should then display the 3 2 matrix A = 1 3 5 2 4 6
(MATLAB displays matrices without braces). You could also generate this matrix by pressing the Enter key at the end of each row, instead of typing a semicolon. Now use MATLAB to create the following matrix, row vector and column vector: 1 2 3 1 B= 4 5 6 x= 4 3 2 X= 2 7 8 9 3 Next, type the names of each of these matrices and vectors that you have created at the MATLAB prompt. Note that x and X are dierent objects; MATLAB is case sensitive. Finally, type whos at the prompt to get a list of all the matrices and vectors that are in your current MATLAB workspace. (b) The MATLAB size command determines the number of rows and columns in a matrix. Every MATLAB command is documented in a help le, which you can access during a MATLAB session. Type help size now to get information about this command. Then use the size command to create a 4 2 matrix whose rows are the sizes of A, B, X, x (in that order), by typing [size(A); size(B); size(X); size(x)] Give this matrix the name S by typing S = ans. Note that all matrices which occur in MATLAB must have names; if a matrix is unnamed, then it gets assigned the temporary name of ans at the moment that it is created. (c) To access a given entry in a matrix, put the row and column number in parentheses following the matrix name. Type a32 = A(3,2) and check that a32 is the (3, 2) entry in A dened above. Observe that the equal sign = in MATLAB (as in other programming languages) executes a substitution : the current value of the variable on the right side of the equal sign is placed into the location whose name is on the left side. Type A(3,2) = 7 and check that the (3, 2) entry of A is now 7. Now change the (3, 2) entry of A back to 6. One way to do this is to type A(3,2) = 6. Another way that you should try for future use is
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the up-arrow key . This lets you cycle through the commands that you have already typed. When you get to the command that generated A, press Enter. If you go too far with the up-arrow, you can use the down-arrow key . (d) To access a whole row or column of a matrix, use the colon operator. For example, A(:, 2) is the second column of A, while B(1, :) is the rst row of B. Type C(:,1) = B(:,1); C(:,2)= B(:,3) to create a 3 2 matrix C whose rst column is the rst column of B and whose second column is the third column of B. Then use the colon operator to create a 2 3 matrix D whose rst row is the rst row of B and whose second row is the third row of B. Use MATLAB to display the matrices C and D by typing C, D <Enter> .
Question Block 2. Matrices and Special Matrices
With MATLAB it is easy to form new matrices from those that are already in the workspace and to create matrices of special types. (a) You can create block matrices by putting two matrices side by side (if they have the same number of rows), or one on top of the other (if they have the same number of columns). Use the matrices A, B, C, D, X created in Question 1 to make the following block matrices (the semicolon means that the matrices are stacked one on top of the other). Before typing the MATLAB commands, insert a comment line that lists which of the matrix combinations in this list t together. Then use MATLAB (you will get error messages when the matrix sizes are not compatible). [A X] [B C] [C D] [C; B] [B; D]
(b) Type each of the following commands that generate special matrices. eye(4) zeros(3) zeros(3,5) ones(2,3) diag([4 5 6 7])
(c) The MATLAB command rand(m, n) creates an m n matrix whose entries are random numbers between 0 and 1. Try this by typing R = rand(2, 3). Then use the up-arrow key to generate two more samples of the random matrix R. Notice how the entries in R change each time the command is executed.
More about MATLAB:
Suppressing Displays: When you place a semicolon at the end of a command, the command will be executed but the result will not be displayed on the screen. This is very useful when you are creating big matrices. Try typing z = [3:0.2:5]; to create a long row vector with equally spaced entries(note the semicolon at the end). Then type z to display the vector z. Format Commands: You can control how numbers are displayed on the screen (this does not aect the internal arithmetic). Type y = [4/3 1.2345e-6];
Then observe the values you get for y when you type the following: format format format format short, y long, y short e, y rat, y format short
Before continuing with the lab, type
Question 3. Matrix Addition and Matrix-Vector Multiplication
Be sure that you have set the seed of the random number generator as described at the beginning of this assignment. Generate vectors u and v and matrices A and B by typing
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MATLAB Assignment #1 v = x(10*rand(3,1)), A = x(10*rand(2,3)), B = x(10*rand(2,3))
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u = x(10*rand(3,1)),
These matrices have entries randomly selected from the numbers 0, 1, . . . , 9. (a) To obtain a linear combination sA + tB of the matrices A and B (where s and t are real numbers) using MATLAB, you must type s*A + t*B. This is only dened when A and B are the same size. The transpose AT of A is obtained by typing A . Calculate the following using MATLAB. A + B, B + A, 6B, 2(3B), 6A + 15B, 3(2A + 5B), 3A, (3AT )T
Insert comment lines in your diary le that explain the properties of matrix addition and scalar multiplication that these calculations illustrate (see Theorem 1.1 on page 5 of the text). (b) To obtain the matrix-vector product Au using MATLAB, you must type A*u (remember that a vector is just a matrix with one column). Calculate the following using MATLAB. Au + Av, A(u + v), (A + B)u, Au + Bu, A(3u), 3A(u)
Insert comment lines that explain the properties of matrix-vector products that these calculations illustrate (see Theorem 1.2 on page 20 of the text).
Question 4. Gaussian Elimination and Reduced Row-Echelon Form
Now that you know how to do matrix calculations with MATLAB, you can easily carry out the steps in Gaussian elimination. Before starting work on this question, type rrefmovie at the MATLAB prompt. You will see a step-by-step example of the row operations that transform a matrix A into its reduced row echelon form R = rref(A). In this demonstration, each pivot is chosen to be the largest in its column (for numerical stability), so extra row interchanges are used. Since rref(A) is uniquely determined by A, this does not aect the nal answer. (Do not include this part in your lab write-up.) (a) Generate a 3 4 matrix A with random integer entries by the command A = x(10*rand(3,4)) To transform A into R = rref(A), start with R = A. Normalize the rst row of R to get R(1, 1) = 1: R = A; R(1,:) = R(1,:)/R(1,1) (note the use of the colon to operate on whole rows of the matrix). If your random matrix happens to have A(1, 1) = 0, then interchange rows to get a nonzero entry in the (1, 1) position before doing the calculation above. Now subtract a multiple of the rst row of R from the second row to make R(2, 1) = 0: R(2,:) = R(2,:) - R(2,1)*R(1,:) Repeat this procedure to make R(3, 1) = 0: R(3,:) = R(3,:) - R(3,1)*R(1,:) (you can minimize typing by using the up-arrow key and editing the command line). The rst column of your matrix R should now be the same as the rst column of rref(A). (b) Operate on R by the same method as in (a) to obtain the second column of rref(A). First normalize row 2 of R, then subtract multiples of row 2 from rows 1 and 3 to put zeros in the (1, 2) and (3, 2) positions (you can minimize typing by using the up-arrow key and editing the commands used in part (a)). (c) Operate on R by the same method as in (b) to obtain the third column of rref(A). First normalize row 3 of R, then subtract multiples of row 3 from rows 1 and 2 to put zeros in the (1, 3) and (2, 3) positions. (d) Your matrix R should now be transformed into rref(A) (since A is a random 3 4 matrix, rref(A) is (almost) sure to have rank 3). Check your answer by MATLAB with the command rref(A). If the MATLAB answer is not the same as the current value of your R, go back and redo your calculations.
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Optional Extra-Credit Problem
Read Section 1.2, Example 3 (pages 18-19) in the text. (a) Use MATLAB to solve Exercise #33 (page 22) part (a). Show the MATLAB computations and write comments explaining your solution. (b) Use MATLAB to solve Exercise #33 (page 22) part (b). Show the MATLAB computations and write comments. (c) Use MATLAB to solve Exercise #54 (page 22). Show the MATLAB computations and write comments. Final Editing of Lab Write-up: After you have worked through all the parts of the lab assignment, you will need to edit your diary le. Remove all errors and other material that is not directly related to the questions. In particular, remove from your diary le any of the results generated by the load, save, clear, format commands that you used. Your write-up should only contain the results of the MATLAB calculations and the answers to the questions that you have written. Preview the document before printing and remove unnecessary page breaks and blank space. Put your name, section number, and student ID number on each page. (If you have diculty doing this using your text editor, you can write this information by hand after printing the report.)
Important: An unedited diary le without comments submitted as a lab writeup will get a GRADE OF ZERO.