The following problems are designed to prepare you for Lab 2 on linear algebra.
If you have your own computer and want to run Matlab on it, download and install Matlab from UNH academic software. Otherwise you can do your homework on the computers in Kingsbury N129. Save your Matlab session to a file by turning “diary on”. When you're done, edit the diary in a text editor to remove errors and add appropriate comments. Print the edited diary to turn in. Your name, the date, and “Math 445 HW 1” should appear prominently at the top.
Remember, you are required to do these homeworks by yourself. If you get stuck, refer to your textbook. If you are absolutely stuck, talk to a fellow student, the teaching assistant, or the instructor, and get hints for how to proceed. But do not copy someone else's solved problem.
Problem 1: Given two numeric variables and , write a Matlab expression that evaluates to true (1) if and have opposite signs and false (0) otherwise. By opposite signs, I mean one is positive and one is negative. Test the expression by evaluating it with the following pairs of numbers (x,y) = (-5, 4), (5,4), (5,-4), (0,-2), and (3,0).
Problem 2: The combined resistance of three resistors in parallel is given by
Create variables for the three resistors , with values 5, 3, and 4. Then calculate by translating the above formula into Matlab syntax.
(a) Create a row vector x whose elements are the numbers 5, 7, 10, 1.
(b) Create a column vector x whose elements are the numbers 5, 7, 10, 1.
(c) Use colon syntax to create a row vector x whose elements start at 0, end at 1, and increase in steps of 0.1.
(d) Determine the dimension of x from (c) and assign the value to the variable d (using Matlab, not by counting!).
(e) Use the
linspace function to create a 10-dimensional vector of numbers evenly spaced between 0 and 1.
(a) Create the matrix A
(b) Change to 7.
(c) Assign the third column of A to the variable v.
(d) Change the first row of A to 8, 1, 4
Problem 5: In Matlab, create a 2 x 2 matrix and a 2-d vector . What vector satisfies ?
Problem 6: Use Matlab to solve the problem. Nilanjana has 40 coins worth $6.40. They're all quarters and nickels. How many nickels and how many quarters does she have? Verify that your answer solves the problem.
Hints: Convert the story problem into two equations in two unknowns. Then rewrite this system of equations in matrix-vector notation, , where are the known coefficients of the linear equations, a vector of unknowns, and a vector of known constants. Enter the matrix and the known vector into Matlab, then solve for using Matlab's backslash operator:
x = A\b.
Problem 7: Use Matlab to solve the problem. Suhasini has 44 coins worth $7.50. They're all quarter, dimes, and nickels. She has twice as many dimes as nickels. How many of each type of coin does she have? Find the answer, and then verify that the solution satisfies the problem.