### 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:
- The string of numbers:
5, 2, 25, 11, -2, -7, -15, -15, 32, 14, -8, -13, 38, 19, -19, -19, 37, 16
- The last word of the decoded message: _SUE
- 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 R^{3}
(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)