MTH 285 Graded Assignment 1 (Modules 1-3)

MTH 285 Graded Assignment 1 (modules 1-3) Unless specified otherwise, please use SciLab to do any matrix computations where possible. Be sure to incorporate your SciLab code into your solutions; that's your supporting work, and if you have an incorrect answer I can see if it's as simple as a typo. I'd prefer that you not print the screen directly from SciLab, but take the time to copy and paste the relevant bits into a document, and format everything neatly, so that I know (1) what the question is, (2) what the work is, and (3) what the solution is.

Problem 1:

By “hand” (show step-by-step reduction; in particular, which row operations are being used on each step?), solve the following systems of equations. Use Gauss-Jordan elimination (zeros in both lower and upper triangles) and write the matrix in reduced row echelon form. If the system has no solution, indicate so. If the system has infinitely many solutions, be sure to write the solutions using free parameters.

The “Linear Algebra Toolkit>>Row Operation Calculator” ( http://www.math.odu.edu/~bogacki/lat/ ) is your friend here – you can use it to step through the row operation calculations. These problems are completely checkable, and there’s no excuse for making arithmetic errors! Numbers may in fact be deliberately ugly to strongly encourage checking and effective technology use.

a) System one

b) System two Problem 2:

Consider the system of linear equations

(a) For which right hand sides does the system have no solution? Explain how you determined this.

(b) For which right hand sides does the system have infinitely many solutions? Explain how you determined this.

(c) In the case(s) of infinitely many solutions, write the solution(s) to the system(s) using parameters.

(d) Is there any possible right hand side for which this system has a unique solution? If yes, give the values of a possible . If no, explain why not.

This is a multiple right hand sides problem; glue it together into one big matrix and use SciLab to rref. Note that in SciLab, a value of D-16 (or D-17) indicates power – it’s effectively zero with rounding error and should be read as exactly zero. Problem 3:

Consider the linear system

Here, and are the variables of the system; is a coefficient.

a) Write the system as an augmented matrix, and begin to put it in reduced row echelon form. Perform the steps needed to (1) obtain a “1” in the pivot position in the first row, and (2) produce a “0” in first column of the second row. Stop and record that matrix.

b) Now, analyze the above before going any further:

i. What value of will cause the system to have infinitely many solutions?

ii. What value of will cause the system to have no solution?

c) For all other values of excluding the two you just found above, the system will have a unique solution. Assume not equal to either of those two values, and keep going with the Gauss Jordan elimination. Write the solution to the system with and in terms of the parameter .

Problem 4:

Find the expression for a polynomial of degree 5 that satisfies the conditions

Plot the function and the points to check, and observe that the derivative values make sense (Winplot is nice, your graphing calculator and a good hand sketch on graph paper are acceptable as well). Problem 5:

Solve the circuit problem shown below for the unknown currents. (Once it's set up, you can use SciLab to reduce - no need to do by hand). Start by assuming the directions shown on the diagram; indicate at the end if any of the current directions need to be reversed.