Report Linear Algebra.pdf

TOPIC 1Write a program in Matlab or Python or any software to decode a secret message, namely, your program allows user input the following information: The encoded message, the matrix k

ĐẠI HỌC QUỐC GIA TRƯỜNG ĐẠI HỌC BÁCH KHOA TP.HCM

REPORT LINEAR ALGEBRA

INSTRUCTURE: HOÀNG HẢI HÀ

TOPIC 1 3-6 Solution 3 Python’s code 3-5 Python’s result 5-6

TOPIC 2 7-10 Solution 7-9 Result 10

TOPIC 3 11-18 Solution 11-13 Matlab code 14-15 Matlab’s result 16-18

TOPIC 4 19-25 Solution 15-24 Result 25

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TOPIC 1

Write a program in Matlab or Python (or any software) to decode a secret message, namely, your program allows user input the following information: The encoded message, the matrix key and then your output is the meaningful message

1 SOLUTION:

To decode the message, we take the string of encoded message in numbers andmultiply it by the inverse of the matrix that was used to encode the message to getthe original string of numbers

Finally, by associating the numbers with their corresponding letters, we obtain theoriginal message

In this topic, user input the encoded messasge and the matrix key

the inverse of the matrix to get the original string of numbers

Finally, by associating the numbers with their corresponding letters, we obtain theoriginal message

2 PYTHON CODE:

Use PYTHON determine the decode message with the encoded message and thematrix key entered by user

import numpy as np

from matrix import Matrix

from encoder import Encoder

from decoder import Decoder

# For user input

encoded_message = str(input("Input the encoded message in integer format :"))encoded_message = "[" + encoded_message + "]"

print("Encoded message in integer format : " + encoded_message)

print("Input the matrix key :")

R = int(input("Enter the number of rows:"))

C = int(input("Enter the number of columns:"))

# For printing the matrix

print("The matrix key is:")

Using PYTHON determine the decode message with the encoded messageand the matrix key entered by user :

TOPIC 2

Problem: “A mouse trap is placed in room 1 of the house with the pictured floor

plan Each time the mouse comes into room 1, he is trapped with probability p =0.1 If he is not trapped, he leaves each room by one of its exits, chosen atrandom.”

 Find the matrix transition describing the path of the mouse through thehouse

 A vector q is called steady vector of Markov model if P q = q where P ismatrix transition, find q

 Suppose that the mouse starts in Room 4, what is the probability that themouse in Room 1 after 3 steps?

 Suppose that the mouse starts in Room 5, find the number of expected stepsthat the mouse in Room 4 for the first time

in the room.

When the mouse comes to Room 1, there are 2 possible possibilities: the mousecontinues to travel to other rooms with the probability of 0.9 or being trapped withthe probability of 0.1

The transition matrix is the transition probability data we can generate using theinformation provided

The table below present the transition probability of the mouse in the house

Thus, we have the transition matrix from the table above:

q q

By looking at the powers of the matrix P, we may examine the evolution of theMarkov chain through time (successive transitions) The k-step transition matrix,

So that, we get the transition matrix or the probability for the mouse to reach any

TOPIC 3

Consider an open production model having n industries with n > 1 In order to produce $1 of its own output, the j th industry must spend $(1/n) for the output of the i th industry (for all i 6= j ), but the j th industry (for all j = 1, 2, 3, , n ) spends nothing for its own output Write a program with n is input by a user to do the following

• Construct the input-output matrix C n

• The matrix input-output C is called productive if (I − C) exists and (I − C) ≥ −1 −1

0 Show that C is productive n

• Find the vector output-values of this system.

1 SOLUTION:

By using the given data of the consumption matrix , we can constructed as:

The sum’s entries of each row or each column of is:

We can conclude from here that the consumption matrix is productive By usingMATLAB command inv-inverse, we can determine the inverse of for For weuse command inv( eye(2) ) and this gives us:

Using the same idea for , , we have:

- inv( eye(4) ):

- inv( eye(5) ):

Taking from respectively, we can obtain that the matrix term is in form where the diagonal entries are 2 and all other entries are 1 Therefore, from this astructure if the matrices we gave the following conjecture

The general form of is:

Proving this conjecture, we show that

We have:

Proving our conjecture:

Proof:

Farther detail:

Find the vector output-values of this system:

According to the theory, we have vector output-values equals

is the demand vector

: is the vector output-values

end

outside demand',i);

Using MATLAB to construct the input-output matrix and show that isproductive:

TOPIC 4:

According to the table, we have three possible genotypes AA, Aa, and aa We can now from request that the farmer dersires to undertake a breeding program in which each plant is always fertilized with a plant of genotype AA

SOLUTION:

First of all, we will demonstrate our old knowledge which not related to Linear Algebra which is biology we studied at high school

Case 1 and 2

Case 3 and 4

Case 5 and 6

Result from these equation:

By using the statics in the table 1 and information in the Howard Anton, Chris

Rorres – Linear Algebra 11 edition, th we have:

x(n) = Mx(n−1) From here, we can conclude that:

a = a + b1 0 0 b1 = c + b0 0 c = 01

Thirdly, to derive an expression for the distribution of the three possible genotypes

in the population after n generations in terms of the fractions present after n – 1 by

Continue, we using the same idea in the text book to find eigenvectors

 THE END 

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