Problem Set 3

Problem Set Guidelines

Problem 1: 2.1 # 12

Problem 2: 2.1 #16 e)

Problem 3: 2.2 # 18 (be sure to show the steps and reasons used to obtain A as a part of your explanation)

Problem 4: 2.3 #12 a, b , c and e [note that in b the problem says into which means inside of, just like in English]

Problem 5 in Maple:
2.3 # 41 (with the following instructions):
Use the inverse method in Maple to solve the equations - convert to fractions while doing so, and then only at the end use an evalf command to obtain the decimal values of the solutions.
To find the percent error for each x_i, look at the magnitude of the difference and divide by the value of x_i in the first system.

Problem 6 in Maple:
Assume that you intercept a number of items, as follows:
  1. The string of numbers: 35, 10, 42, 21, 5, 2, 28, 14, 17, -2, 3, 0, 5, 0, 8, 4, 11, 3, 31, 13
  2. The last word of the decoded message: HERE
  3. The fact that 2x2 matrix was used in the Hill Cipher
Can the rest of the message be decoded? If not explain why not, and if so, show by-hand and/or Maple work to obtain the decoding matrix and the decoded message.

I'll be posting responses to select ASULearn messages I receive from the class in the forum on ASULearn - so look there for hints and suggestions.
A Review of Various Maple Commands:
> with(LinearAlgebra): with(plots):
> A:=Matrix([[-1,2,1,-1],[2,4,-7,-8],[4,7,-3,3]]);
> ReducedRowEchelonForm(A);
> GaussianElimination(A); (only for augmented matrices with unknown variables like k or a, b, c in the augmented matrix)
> ConditionNumber(A); (only for square matrices
> Transpose(A)
> Vector([1,2,3]);
> B:=MatrixInverse(A);
> A.B;
> A+B;
> B-A;
> 3*A;
> A^3;
> evalf(M)
> spacecurve({[4*t,7*t,3*t,t=0..1],[-1*t,2*t,6*t,t=0..1]},color=red, thickness=2); plot vectors as line segments in R3 (columns of matrices) to show whether the the columns are in the same plane, etc.
> 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); plot equations of planes in R^3 (rows of augmented matrices) to look at the geometry of the intersection of the rows (ie 3 planes intersect in a point, a line, a plane, or no common points)