### 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:
- The string of numbers:
35, 10, 42, 21, 5, 2, 28, 14, 17, -2, 3, 0, 5, 0, 8, 4, 11, 3, 31, 13
- The last word of the decoded message: HERE
- 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 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)