### Problem Set 3

See the Guidelines. I will post on ASULearn answers to select questions I receive via messaging there or in office hours.

Problem 1:  2.1 # 12 with additional instructions:
Part A: Answer the question as in the book and show your reasoning for how you chose B
Part B: Describe the set of matrix solutions to AB=0 - ie describe all the possible Bs you could have taken.
Part C: Could you have found a B so that it's columns are linearly independent? Show work and explain why or why not.

Problem 2:  2.1 # 22 using both matrix multiplication and associativity, as well as the definition of not l.i., in your reasoning

Problem 3:  2.3 # 41 with additional instructions: Solve the book problem, but 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.
Note: The condition number command is listed below in the Maple commands

Problem 4:
Assume that you intercept a number of items, as follows:
1. The string of numbers: 5, 2, 25, 11, -2, -7, -15, -15, 32, 14, -8, -13, 38, 19, -19, -19, 37, 16
2. The last word of the decoded message: _SUE
3. The fact that 2x2 matrix was used in the Hill Cipher
We'll investigate whether the rest of the message can be decoded as follows:

Part A: Multiply two matrix vector equations for a decoding matrix, either by-hand or in Maple:
DecodingMatrix:=Matrix([[a, b],[c,d]]);
DecodingMatrix.Vector([-19,-19]) = Vector([0,19]);
DecodingMatrix.Vector([37,16]) =
... (the vector corresponding to UE)

Part B: From setting equal each corresponding entry in Part A, write down the 4 equations in the 4 unknowns a, b, c, d.

Part C: Solve this system (it's linear - you can solve it using a variety of methods, like a 4x5 augmented matrix with columns a b c d and an equals column, so your first row would be [-19 -19 0 0 0]) to see whether you have 0, 1 or infinite solutions for a, b, c and d.

Part D: If you have solutions, put them back into DecodingMatrix:=Matrix([[a, b],[c,d]]); and use this to decode the
CodedMessage:=Matrix([[5, 25, -2, -15, 32, -8, 38, -19, 37],[2, 11, -7, -15, 14, -13, 19, -19, 16]]);
by performing DecodingMatrix.CodedMessage, and then translate back to letters. If not, then explain why the system is inconsistent.
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);
decimal approximation of 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)