Dr. Sarah's Maple Commands and Hints for Problem Set 1

I will post on ASULearn answers to select questions I receive via messaging or in office hours.Be sure to carefully follow the guidelines and both the online and book directions in order to receive full credit. Review Exercise Solutions that are on ASULearn, which will often help you with problem set problems.

Problems 1 and 4 require you to use and print your Maple work. Sample commands follow at the end of this hint sheet. For the by-hand reduction work in Problem 1, be sure to provide the row operation instructions and show work.

The problems require you to reason (and explain your reasoning) of solutions. Recall that as long as there is not any inconsistent row in Gaussian, like

[0 0 0 non-zero]

which gives inconsistent because 0 times variables can't be non-zero, then the missing pivots, if any, give you free variables, like x3=s and x5=t to use to write out the parametric equation, if there are no pivots in those spots. If there is not a pivot in the augmented spot, then full pivots of the variables provide a unique solution. You can use geometry for your justification, like parallel planes, etc, as well as the algebra of examining rows like

[0 0 nonzero anything]

which gives 1 solution for x3, because you could solve for x3=anything/nonzero, which is a specific number

Use Gaussian Elimination instead of Gauss-Jordan on problems with unknown variables, like Problem 4. We use ReducedRowEchelonForm [Gauss-Jordan] on problems with all numbers in them.

For Problem 3, be careful when you write out the general solution - think carefully about how many free variables there are and which of the variables can be free (ie the others depend on these). In this case, after reduction, there is no pivot in the equals column (which would tell us inconsistency) and there are 3 non-trivial equations. Since there are 5 variables, that must mean that 2 of them are free (s and t). Those variables without a pivot/where a leading number corresponds to that column spot can be set as free.

For true/false type statements like Problem 2 and 5, if the statement is true, quote the general statement and page number from the book that tells you so. If it is false, find one specific instance and example - it does not suffice to argue that the book says something slightly different which does not match up with the statement (as both statements could be true), nor to claim you could find a counerexample - you must actually produce one. Try to find simple counterexamples like a system with 2 unknowns, and explain why your example violates the original statement.

Here are some commands that will be useful. Be sure to also see the problem set guidelines. For example, a problem in Maple must utilize Maple text comments.

Execute at the Start of Every 2240 Maple

`> `
**with(plots): with(LinearAlgebra):**

Plotting a System of Equations

`> `
**implicitplot({2*x+4*y-2,5*x-3*y-1}, x=-1..1, y=-1..1);**

`> `
**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);**

Augmented Matrix and Gauss-Jordan (or Reduced Row Echelon) Form

`> `
**A:=Matrix([[-1,2,1,-1],[2,4,-7,-8],[4,7,-3,3]]); ReducedRowEchelonForm(A);**

Hence we see that the solutions are x=195/83, y=-15/83, z=142/83.

Gaussian (or Row Echelon) Form

Note that if there are unknown variables in the matrix, you should use GaussianElimination instead.

`> `
**P:=Matrix([[1,3,4,k],[2,8,9,0],[10,10,10,5],[5,5,5,5]]); GaussianElimination(P);**

Analysis will show that this is impossible - the 4th row tells us 0x+0y+0z=5/2, which is impossible.