### 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:
1. A question from sender a: You walk into the linear algebra petting zoo. Who do you see?
2. 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
3. The beginning letters of the decoded message: Shea
4. 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 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)