### 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.