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 # 22 using both matrix multiplication and associativity, as well as the definition of not l.i. in matrix vector form (Bx=0 has...), in your reasoning. Be careful: B and AB are not necessarily square here, so don't assume they are--ie you can't apply 2.3 material.

Problem 2:  2.2 # 18: Show work, annotate your reasoning, and use associativity in your reasoning.

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))

Problem 4 Hill Cipher 
Assume that you intercept a number of items, as follows:
  1. A question from sender a: What does the noninvertible matrix suffer from?
  2. A string of coded numbers as a reply: -12, 13, -9, 9, 3, 1, 8, 6, -2, 11, 50, -25, -12, 15, -1, 10, -1, 10
  3. The first 4 letters of the decoded message: An_i
  4. 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([-12, 13]) = Vector([1, 14]); (the vectors corresponding to An)
DecodingMatrix.Vector([ fill in, fill in]) = Vector([ fill in , fill in ]);... (the vectors corresponding to _i - 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 [-12 13 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

CodedMessage:=Matrix([[-12, -9, 3, 8, -2, 50, -12, -1, -1],[13, 9, 1, 6, 11, -25, 15, 10, 10]]);
DecodingMatrix.CodedMessage;

Be sure to define the DecodingMatrix correctly using your work in Part C, with a, b as the first row. Notice also that the coded string has gone in correctly 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 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)