### 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 # 26 with additional instructions:

**Parts a and b**:
Answer the questions as in the book (note there are two of them)
and show your reasoning. **Be careful**: A is not necessarily square
here, so don't assume it is--ie you can't apply 2.3 material.
Show work, annotate your reasoning, and use associativity in your reasoning.

**Part c**: What does this tell you about whether the columns of A span
the entire space they lie in? Explain and include the formal definition of
span in your annotations.

**Problem 2:**
2.2 # 14:
Show work, annotate your reasoning, and use associativity in your reasoning.

**Problem 3:**
2.3 # 41 with additional instructions:

**Part a**:
Follow the books instructions for part a,
but solve these systems and find the exact solution using fractions and
the inverse method, using evalf only at the end. Here are the commands
for three decimals. Execute these and then modify b to compare with what
happens using two decimals.

M := Matrix([[45/10, 31/10], [16/10, 11/10]]); P := MatrixInverse(M);

bthreedecimals := P.Vector([[19249/1000, 6843/1000]]);

evalf(bthreedecimals);

**Part b**: Follow the books instructions for part b.
As listed there,
to find the percent error for each coordinate (x_1 and then x_2)
look at the magnitude of the
difference between the values |xthree_i -xtwo_i|
and divide by the value from the three decimal system (xthree_i).
Show work.

**Part c**: As in the book, find

ConditionNumber(M);

**Part d**:
The condition number gives a rough measure of how sensitive the solution of
Ax=b can be to changes in b. The condition number measures the
asymptotically worst case of how much the function can change in proportion
to small changes in the argument.
As a general rule of thumb, if the condition number has order
~10^k then we may lose up to k digits of accuracy.
What is the order of the condition number here?

**Problem 4** Hill Cipher

Assume that you intercept a number of items, as follows:
- A question from sender a:
*What do you get when you take the dot product of a vector and a mountain
climber?*
- A string of coded numbers as a reply:
-4, 17, 1, 14, 0, 19, 7, -5, 7, -2, -3, 6, 3, 6, 11, -9, -13, 31,
19, -19, -17, 35, 5, -5, 16, -13, -11, 23, -17, 35, 19, -19
- The last 4 letters of the decoded message: ARS_
- The fact that a 2x2 matrix was used in the Hill Cipher

We'll investigate whether the rest of the message can be decoded as follows:

**Be sure to annotate your work!**

**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([-17, 35]) = Vector([1, 18]);
(the vectors corresponding to AR)

DecodingMatrix.Vector([ fill in, fill in])
= Vector([ fill in , fill in ]);... (the vector corresponding to
S_ - you fill in)

**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 [-17 35 0 0 1])
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]]); (careful about which numbers are in which spots) and use this to decode.
In Maple, execute (twice!)

interface(rtablesize=16);

interface(rtablesize=16);

CodedMessage:=Matrix([[-4,1,0,7,7,-3,3,11,-13,19,-17,5,16,-11,-17,19],[17,14,19,-5,-2,6,6,-9,31,-19,35,-5,-13,23,35,-19]]);

DecodingMatrix.CodedMessage;

The interface command gets Maple to display more than 10 columns of a matrix and you must define the DecodingMatrix correctly using Part C. Notice also that the coded string has gone in as the column vectors of the CodedMessage.

Then translate back to letters (get it?) or 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)