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Name: Applications Assignment 2

1. (10pts)Review“PolynomialInterpolation”in1.9onpages91-93.Thegraphofaquadratic polynomial passes through the points (1, 4), (2, 0), and (3, 12). (a) Write down a system of linear equations that represents the coordinate equations. (b) Use Cramer’s Rule to solve the system.

(c) Give the quadratic polynomial whose graph passes through the points (1,4), (2,0), and (3, 12).

2. (10 pts) Cryptography. The matrix B contains a message in numeric format, where 0=space, 1=a, 2=b, . . ., 25=y,

and 26=z. A simple substitution cipher can often be cracked by looking at the frequency of the digits. Multiplying B by another matrix helps to defeat attacks by frequency analysis.

2 1 1 33 47 14 27 64 41 20

A=1 0 1 AB=11 27 9 1 29 20 10

1 1 1 26 29 9 26 44 41 10

(a) Use the adjoint formula to find A−1. (b) Explain how to use A−1 to calculate B. To make the actual calculation, you may use a calculator or computer algebra system. (c) What is the message contained in B? You may add punctuation as appropriate.

3. (20 pts) Row space and solution space. For this problem, you may use a calculator or computer algebra system to graph the spaces,

but you must include a sketch or printout with the graphs labeled. For each matrix M, do the following:

(a) Graph the row space of M. (b) Find the reduced row echelon form of M. (c) Graph the row space of the reduced row echelon form of M. (d) Find all solutions to the system M⃗x = ⃗0. Give these solutions as both a vector equation and as a set of parametric equations. (e) GraphthesolutionspaceofM⃗x=⃗0. (f) Combine all three graphs from parts (a), (c), and (e) on one set of axes.

123 C=

101

1 2 −1 D=

2 4 −2