Problem Set 1

See the Guidelines 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:  1.1 # 14 using three methods (don't forget to annotate and to solve for the solutions like the book asks for):
Part a) by-hand Gaussian
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 part 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 #25 using Gaussian in Maple and reason from there.
Part b) Use your equation from part a) to choose values for g, h, and k so that the system is consistent (i.e. 1 or infinite solutions). Using the method of your choice produce the solutions in point or parametric form for this choice of g, h, and k.
Part c) Use ReducedRowEchelonForm in Maple on the original matrix in #25 with g, h, and k left as general and show that Maple gives an incorrect answer (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 of backwards/total = backwards / (forwards + backwards) for n=20 and n=200 using the numerical note on page 20, and give decimal approximations too. Note that the forward phase is Gaussian, and the backward phase is from Gaussian to Gauss-Jordan.
Part b) Is the ratio increasing, decreasing or staying constant? Interpret what this is telling you in the language of Gaussian/Gauss-Jordan.
Part c) 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? (Hint: use part a) with n=20 and compare f(n) and f(10n) to see if f(10n) = 10y, where f is the ratio)