**This is an old revision of the document!** ----
====== Math 445 HW1 ====== 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 {{http://at.unh.edu/acs/services/software/ | 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. Remember, you are required to do these homeworks **by yourself**. **Problem 1:** Given two numeric variables $x$ and $y$, write a Matlab expression that evaluates to true (1) if $x$ and $y$ 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 $R_T$ of three resistors $R_1, R_2, R_3$ in parallel is given by \begin{eqnarray*} R_T = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}} \end{eqnarray*} Create variables for the three resistors $R_1, R_2, R_3$, with values 5, 3, and 4. Then calculate $R_T$ by translating the above formula into Matlab syntax. ** Problem 3:** \\ **(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. ** Problem 4:** \\ **(a)** Create the matrix A \begin{eqnarray*} A = \left[ \begin{array}{rrr} 3 & 9 & 2 \\ -1 & 4 & 6 \\ 5 & -2 & 0 \end{array} \right] \end{eqnarray*} **(b)** Change $A_{2,3}$ 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 <latex> A = \left[ \begin{array}{cc} 4 & 2 \\ -1 & 5 \end{array} \right] </latex> and a 2-d vector <latex> b = \left[ \begin{array}{cc} 3 \\ 4 \end{array} \right] </latex>. What vector $x$ satisfies $Ax=b$? **Problem 3:** 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, $Ax=b$, where $A$ are the known coefficients of the linear equations, $x$ a vector of unknowns, and $b$ a vector of known constants. Enter the matrix $A$ and the known vector $b$ into Matlab, then solve for $x$ using Matlab's backslash operator: ''x = A\b''. **Problem 4:** Use Matlab to solve the problem. Suhasini has 44 coins worth $7.45. 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?
