### Problem Set 3

See the Guidelines. 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 # 12 with additional instructions:
Part A: Answer the question as in the book and show your reasoning for how you chose B
Part B: Could you have found a B so that its columns are linearly independent? Explain why or why not.

Problem 2:  2.1 # 26 with additional instructions:
Caution: A and D are not necessarily square matrices so Theorem 8 in 2.3 does not apply!
Part A: Show that for any b... (as directed in the book). Apply the hint in the book and use associativity as part of your reasoning.
Part B: Explain why A cannot have more rows than columns (as directed in the book).
Part C: Write the definition of the span of a set of vectors. Next, what do the columns of A in this problem span?

Problem 3:  2.3 #44 with additional instructions:
Part a: Enter the fifth-order Hilbert matrix into Maple as a matrix with fractions and define it to be
A:=Matrix([[1,1/2,1/3,1/4,1/5],[1/2,1/3,1/4,1/5,1/6],[1/3,1/4,1/5,1/6,1/7],[1/4,1/5,1/6,1/7,1/8],[1/5,1/6,1/7,1/8,1/9]]);
Part b: Compute ConditionNumber(A); in Maple. Write it in scientific notation. What is k?
Part c: In the instructions above 42, substitute r=10, the default # of digits of precision for Maple, and your value of k from Part b - what is r-k?
Part d: As on p. 108 we only need the last column of the inverse, so we can solve Ax=Vector([0,0,0,0,1])]) for x by entering the following commands in Maple
AugmenteddecimalA:=Matrix([evalf(A),Vector([0,0,0,0,1])]);
ReducedRowEchelonForm(AugmenteddecimalA);

Part e: Compute the inverse of A using
MatrixInverse(A);
Part f: For this matrix, is Maple in Part d more, less or the same accuracy as predicted from Part c for the last column of the matrix that is given exactly in Part e (ie compare each entry in the last columns from Part d and Part e. Is each entry the same within the number of digits predicted using part c)? Show work/reasoning... (Note: here 629.998135201517 is accurate rounded to 5 digits (630 rounded) but not 6 (629.998 rounded)--in this way, check the rest)

Problem 4:
Assume that you intercept a number of items, as follows:
1. A question from sender a: Why was the matrix arrested?
2. A string of numbers as a reply from sender b: -11, 23, -9, 18, 24, -36, 8, -11, -9, 19, -5, 10, 22, -30, 29, -40
3. The beginning letters of the decoded message: An i
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([-11,23]) = Vector([1,14]);
(the vector corresponding to the numbers for "a n")
DecodingMatrix.Vector([-9,18]) = ... (the vector corresponding to the numbers for "space i" - 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 [-11,23,0,0,1]) 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 (by-hand or in Maple). If you are using Maple, then execute
CodedMessage:=Matrix([[-11, -9, 24, 8, -9, -5, 22, 29], [23, 18, -36, -11, 19, 10, -30, -40]]);
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: (Individual Video)
Create an individual video (not in groups) 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)