ps1hints.mws

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

Also see Practice Problem Solutions for 1.1 and 1.2, which are on ASULearn.

Problem 1: 1.1   60 part c
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.

Problem 2: 1.1   74
Use Gaussian Elimination instead of Gauss-Jordan on problems with unknown variables. For 74 just find one set of values that works for each part and justify. You can use geometry for your justification, like parallel planes, etc. See the practice problems (1.1 #73 on ASULearn) as an example.

Problem 3: 1.2   30 by hand and also on Maple

Problem 4: 1.2   32
This is much faster on Maple! Since Maple or by-hand is your choice for this problem, you should choose Maple. Note that for #30 you must use both methods.

Problem 5: 1.2   44 find all values of k and justify
Use Gaussian Elimination instead of Gauss-Jordan on problems with unknown variables. You can use geometry for your justification, like parallel planes, etc. For 44, examine the Gaussian reduced matrix and use it to justify and find all the values of the unknown k. See the practice problems (1.1 #73 on ASULearn) as an example.

Problem 6: 1.3   24 parts a and b
Set up the system and solve. Careful in part a 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 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 row where a leading number corresponds to that spot can be set as free.

Problem 7: 1.3   26


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.

> with(plots): with(LinearAlgebra): 

Warning, the name changecoords has been redefined

Plotting a System of Equations

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

[Maple Plot]

> 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);

[Maple Plot]

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);

A := _rtable[136869008]

_rtable[136871232]

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);

P := _rtable[136875680]

_rtable[136878040]

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