Exploration Assignment #1. (Linear Systems)

Transcription

1 Math 0280 Introduction to Matrices and Linear Algebra Exploration Assignment #1 (Linear Systems) Acknowledgment The MATLAB assignments for Math 0280 were developed by Jonathan Rubin with the help of Matthew Badger SCHEDULE: The first exploration assignment is due within two weeks. One submission per group (no more than three people) is sufficient. Every member of the group assumes full responsibility for the final product and should be prepared to answer any questions related to the work submitted. GOAL: In these exploration assignments, our goal is to introduce you to various applications of linear algebra to real world problems. We will make use of the software package MATLAB, a useful tool for numerical computation using matrices and graphical interpretation of data. The first assignment begins with a gentle introduction to MATLAB. You will learn how to enter and carry out basic computations with matrices. In particular, linear systems of equations can be solved quickly and accurately using the rref command, which performs Gauss-Jordan elimination on a matrix. We also examine two applications: (1) Given n points (x 1, y 1 ),..., (x n, y n ) on the Euclidean plane, we can find a polynomial p(x) = a 0 +a 1 x+ +a n 1 x n 1 going through the points: p(x i ) = y i, for all 1 i n. (2) Using linear systems to calculate unknown flows through the junctions of a network. (For example, think of the speed of water through pipes or vehicle traffic on roads.) SAVING YOUR WORK IN THE GSCC COMPUTER LAB: When you save your MAT- LAB files in a computer in the GSCC lab, make sure you save it in a folder called AFS. Otherwise, you will lose everything when you log off of the computer. You can access this folder by double-clicking the link on the Desktop of any GSCC computer. If you want to access this folder using any other computer, you can open a web browser and in the address bar type: ftp://username:userpassword@unixs.cis.pitt.edu. Here username and userpasswword are the same as your pitt username and password. INTRODUCTION TO MATLAB: MATrix LABratory, or more simply called MATLAB, is an interactive software system for doing calculations with matrices. For those interested, several thorough introductions to MATLAB are available online. Here we will focus on just enough to get you through the first assignment. I. The Command Window Once you open MATLAB, several windows will open. Our primary focus is on the window called the Command Window. This is place where you enter commands and receive output. II. Inputing Matrices and Solving Linear Systems 1

2 2 First, suppose that we want to enter the following augmented matrix, A = Then, at the command window prompt, type the following: PROMPT>> A=[1, 2, 3, 4; 5, 6, 7, 8; 9, 10, 11, 12] <enter> MATLAB has assigned the variable A to the matrix and prints the following output to the command window: A = To summarize, a matrix can be entered into MATLAB, by writing the matrix entries rowwise: put the entries between square brackets [... ], put commas, between the entries, and put semicolons ; between the rows. Second, suppose we want to change the bottom left entry in the matrix A from 9 to 9. Do we have to retype the entire matrix into MATLAB? The answer is no. Instead, just type the following into the command window prompt: PROMPT>> A(3,1) = -9 <enter> Then, MATLAB records the change: A = More generally, to change the ij entry of a matrix M, type M(i,j)=... Third, suppose we want to solve the linear system represented by A, x + 2y + 3z = 4 5x + 6y + 7z = 8 9x + 10y + 11z = 12 MATLAB implements Gauss-Jordan elimination, which reduces a matrix to its reduced row echelon form. The following command assigns the RREF of A to the matrix B: PROMPT>> B=rref(A) <enter> This outputs: B = Fourth, suppose we want to extract the solution vector b from B. In the command window, the following command assigns the fourth column of B to b: PROMPT>> b=b(:,4) <enter> MATLAB responds:

3 b = More generally, to extract the ith row or jth column of a matrix M, type M(i,:) or M(:,j), respectively. III. Evaluating Polynomials and Plotting Graphs Suppose that we want to plot the polynomial P (x) = x on the interval [ 1, 1]. Apply the following procedure. First, we need to represent P (x) as a vector p: PROMPT>> p=[-1;0;3] <enter> More generally, to create a vector for the degree n polynomial P (x) = a n x n + + a 1 x + a 0, type p=[a n ; ; a 1 ; a 0 ]. Second, specify the domain u of the desired plot, as follows: PROMPT>> u=-1:0.2:1 <enter> The command above (-1:0.2:1) creates a vector u with entries between 1 (the left number) and 1 (the right number). The middle number 0.2, which is called the mesh or step size, specifies the detail of the plot. A smaller step size (or finer mesh ) produces a more accurate plot than a larger step size. Third, use the following command to evaluate P (x) on the domain specified by u: PROMPT>> v=polyval(p,u) <enter> Lastly, plot the graph by specifying the domain u and the range v: PROMPT>> plot(u,v) <enter> After entering the command, MATLAB will open a new window with its plot of x You will notice that the graph of x is not smooth; rather, it is a linear approximation using line segments, spaced 0.2 apart on the x-axis. Try repeating the procedure above, except replace 0.2 in step 3 with 0.1. The new plot is a finer approximation of x IV. Using M Files to Save Your Work Rather than type commands into the prompt one by one, MATLAB provides a mechanism for collecting your commands in one place. Let s use the previous section as an example. From the MATLAB menubar click File -> New -> M-File. A new window Editor Untitled will open. Retype the commands from the last section: % Any Line That Begins With A Percent Sign Is A Comment % These Commands Plot -x 2 +3 From -1 to +1 p=[-1;0;3] u=-1:0.2:1 v=polyval(p,u) plot(u,v) Save the M-file as test1.m and close the editor. At the command prompt, you can now type a single command to plot the graph of x 2 + 3: PROMPT>> test1 <enter> Completed assignments are to be submitted on paper, clearly indicating the names of all students in the group. A computer printout is preferred, but fully or partially handwritten submissions will be accepted. 3

4 4 ASSIGNMENT: (OUT OF 40 POINTS) Complete all three parts: I. Exercise on Linear Systems (14 POINTS) (a) Convert the following linear system to an augmented matrix, call it A. x 1 + x 2 + x 3 + x 4 = 0 x 1 + 2x 2 + 2x 3 + 2x 4 = 4 3x 2 + 2x 3 + x 4 = 10 2x 1 + x 4 = 4 (For each part, type the appropriate commands in the M-file.) (b) Find the reduced row echelon form of A, call it B. (c) Does the linear system above have a unique solution? If yes, what is the solution? If no, how do you know? (d) Change the matrix A as follows: increase the value of the (2,1) entry by 1. (e) Find the reduced row echelon form of the new A, call it C. (f) Does the new linear system (matrix A) have a unique solution? If yes, what is the solution? If no, how do you know? (g) Describe the difference between B and C. II. Finding Polynomials Cardiac Hill (14 POINTS) (a) Suppose we want a curve through the points (0,5), (1,3), (3,2). unique quadratic polynomial (degree 2) P (x) = ax 2 + bx + c We will find the whose graph goes through these points. Substituting the values for x and y, we get a linear system 0a + 0b + 1c = 5 1a + 1b + 1c = 3 9a + 3b + 1c = 2 (STOP! Make sure you understand how to get this system from the given points.) Solve for the unknown coefficients a, b, c. (b) Plot the polynomial P (x) = ax 2 + bx + c, with the coefficients from part (a), on the interval [0,3]. Use a step size of 0.1 or smaller. (c) Suppose we want a curve through (0,5), (1,3), (3,2) and (2,10). Since there are now four points, we need a cubic polynomial (degree 3) Q(x) = ax 3 + bx 2 + cx + d. (In general, to go through n points, one needs a polynomial of degree n 1.) Find the unknown coefficients of Q as in part (a). (d) Plot the polynomial Q(x) = ax 3 + bx 2 + cx + d, with the coefficients from part (c), on the interval [0,3]. Use a step size of 0.1 or smaller. (e) Does the plot in part (b) radically differ from the plot in part (d)? Justify your answer.

5 5 Figure 1 (f) A linear algebra student lives in Sutherland Hall. As part of his morning commute to MATH 0280, the student walks 200ft (vertical height) down cardiac hill, on the steps next to the Peterson Events Center. From the top of the hill to the bottom of the events center, there are three sets of stairs and two flats areas in between stairs. The first set of stairs drops 20ft, the first flat area is 100ft long, the second set of stairs drops 100ft, the second flat area is 200ft long, and the final set of stairs drops 80ft. Assuming that the vertical drop of each step equals its horizontal stride, find a polynomial that models cardiac hill. (g) Plot your polynomial from part (f). III. Network Analysis A Traffic Problem (12 POINTS) (a) The University of Pittsburgh in central Oakland is littered with one-way streets. For example, the intersection of Fifth and Forbes Avenues with Bouquet Street and Oakland Avenue is modeled by the Figure 1. The arrows represent the flow of traffic, measured by the average number of vehicles entering and exiting each intersection per minute. The rule of conservation states the the number of vehicles entering an intersection must equal the number of vehicles exiting an intersection. For instance, at the intersection of Forbes and Oakland (A), we get (cars in) = 20 = f 1 + f 2 (cars out). Form a linear system in variables f 1, f 2, f 3, f 4, by writing the relationships for the other three intersections. Write this as an augmented matrix, call it T. (b) Solve the linear system from part (a) in terms of the free variable f 4. (c) If the traffic is regulated on CD so that f 4 = 10, what will the average flows on the other streets be? (d) What is the minimum and maximum possible flows on each street? [Hint: Traffic flow cannot be negative.] (e) Would the solution change if all of the directions were reversed? Briefly explain your reasoning. (NOTE: If so, there is no need to compute the new solution.) (f) Would the solution change if we reversed the direction just along Fifth avenue (BC)? Briefly explain your reasoning. (NOTE: If so, there is no need to compute the new solution.)