### Problem Set 3

See the Guidelines and Maple Tips.
I will post on ASULearn answers to select questions I receive via messaging
there or in office hours. I am always happy to help!

*Mathematics, you see, is not a spectator sport.* [George Polya, *How to Solve it*]

**Problem 1:**
2.1 # 25.

**Part A**: Show *C=D* and show all reasoning and steps.

**Part B**: Show why *m=n*. (Hint: In practice problems
we showed that *CA=I* in 2.1 #23 meant that the columns of *A* were linearly
independent and *A* must have at least as many rows as columns in order to ensure that there are no free variables. You can
extend these arguments here.)

**Problem 2:**
2.2 # 16 and show all reasoning and steps.

**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:**

Assume that you intercept a number of items, as follows:
- A question from sender a:
*You walk into the linear algebra petting zoo. Who do you see?*
- A string of numbers as a reply from sender b:
11, -3, 4, -3, 13, -8, 4, -4, 11, -3, 0, 5, 16, -16, -13, 27, 4, -4, 11, -10, 11, -9, -4, 8, 13, -5
- The beginning letters of the decoded message:
*Shea*
- 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:

**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([11,-3]) = Vector([19,8]); (the vector corresponding to the numbers for "s h")

DecodingMatrix.Vector([4,-3]) = ... (the vector corresponding to
the numbers for "e a" - create it!)

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 ReducedRowEchelonForm of a 4x5 augmented matrix with columns
*a b c d* and an equals column, so your first row would be [11,-3,0,0,19])
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 (by-hand or in Maple). If you are using Maple, then execute (twice!)

interface(rtablesize=13);

interface(rtablesize=13);

CodedMessage:=Matrix([[11,4,13,4,11,0,16,-13,4,11,11,-4,13],[-3,-3,-8,-4,-3,5,-16,27,-4,-10,-9,8,-5]]);

DecodingMatrix.CodedMessage;

You must first 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 by reading down the columns. If there are no solutions, then explain why the system is inconsistent.

**Problem 5:**
Create a video on a first-come, first-served topic. See the link
for more information and instructions.

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)