Problem Set 1

See the Guidelines, Maple Tips, and the Maple Commands/Template for Problem Set 1. I will post on ASULearn answers to select questions I receive via messaging or in office hours. I am always happy to help!

Mathematics, you see, is not a spectator sport. [George Polya, How to Solve it]

Problem 1:  1.2 # 4 using three methods (don't forget to annotate):
Part A: By-hand Gaussian. List the pivots, pivot columns, and solve for the solutions using back substitution.
Part B: ReducedRowEchelonForm in Maple
Part C: implicitplot3d in Maple, and describe what you see and how this connects to the question
Part D: Do all the methods yield the same solution(s)? Compare and contrast.

Note for Parts B and C, you can use commands like the following, but replacing with the coefficients from this question:
with(plots): with(LinearAlgebra):
Pr1:=Matrix([[-1,2,1,-1],[2,4,-7,-8],[4,7,-3,3]]);
ReducedRowEchelonForm(Pr1);
implicitplot3d({x+2*y+3*z=3,2*x-y-4*z=1,x+y+z=2}, x=-4..4,y=-4..4,z=-4..4);

Problem 2:
Part A: 1.1 #27 using Gaussian and reason from there.
Part B: Use the original matrix from Part A but forget about the rest of the text for this part. Choose an example of a, b, c, d, f, g with a still not 0 so that the system has infinite solutions. Write the solutions in parametric form.
Part C: Use ReducedRowEchelonForm in Maple on the matrix with the variables a, b, c, d, f, g all left as general, and then argue from there that Maple gives an incorrect solution(s) for your values in Part B (recall we should only use ReducedRowEchelonForm when the array is all numbers). Next: How many solutions do we obtain here and how is this different from part B?

Problem 3:  1.2 #30 - produce the example and show that your example is inconsistent

Problem 4:  1.2 #32 -
Part A: Compute the exact ratios (don't approximate!) of
backwards/total = backwards / (forwards + backwards)
for n=20 and n=200 using the numerical note on page 20.
Note that the forward phase is Gaussian, and the backward phase is from Gaussian to Gauss-Jordan.
Part B: Then give decimal approximations too.
Part C: Is the ratio increasing, decreasing or staying constant?
Part D: Next interpret what this is telling you in the language of Gaussian/Gauss-Jordan.
Part E: If a function f is linear then when f(n) = y, we know that f(10n) = 10y because the change in y / change in n must be a constant. Is the ratio a linear function of n? To answer this, use Part A with n=20 and compare f(n) and f(10n) to see if f(10n) = 10y, where f is the ratio.