Part A: Answer the question as in the book and show your reasoning for how you chose

Part B: Could you have found a

Caution:

Part A: Show that for any

Part B: Explain why

Part C: Write the definition of the span of a set of vectors. Next, what do the columns of

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]]);

AugmenteddecimalA:=Matrix([evalf(A),Vector([0,0,0,0,1])]);

ReducedRowEchelonForm(AugmenteddecimalA);

MatrixInverse(A);

Assume that you intercept a number of items, as follows:

- A question from sender a: Why was the matrix arrested?
- 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
- The beginning letters of the decoded message:
*An i* - The fact that 2x2 matrix was used in the Hill Cipher

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

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

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.

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:

> A:=Matrix([[-1,2,1,-1],[2,4,-7,-8],[4,7,-3,3]]);

> ReducedRowEchelonForm(A);

> GaussianElimination(A);

> ConditionNumber(A);

> Transpose(A);

> Vector([1,2,3]);

> B:=MatrixInverse(A);

> A.B;

> A+B;

> B-A;

> 3*A;

> A^3;

> evalf(M);

> spacecurve({[4*t,7*t,3*t,t=0..1],[-1*t,2*t,6*t,t=0..1]},color=red, thickness=2);

> 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);