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 # 12 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) 1.2 #20 using Gaussian and reason from there.
Part b) Choose an example of h and for k
so that the system has infinite solutions.
Write the solutions in parametric form.
Part c) Use ReducedRowEchelonForm in Maple on the matrix with h, k left
as
general, and then argue from there that Maple gives
an incorrect solution(s) for your h, k 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 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)