What is it? Students will write a short, formal expository paper about some application of the ideas of linear algebra. This will involve independent reading, thinking, and learning. Students will...

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What is it?
Students will write a short, formal expository paper about some application of the ideas of linear algebra. This will involve independent reading, thinking, and learning. Students will choose a topic which we have not directly covered in class but which is connected to topics we have discussed, study it using the text and possibly other resources, then write expository papers explaining what they have learned.

Objectives.
Students will be graded on a variety of objectives.

  • Organization, grammar, spelling, basic language arts. This is college-level writing, and good thoughts that you don't know how to communicate are worth almost nothing.

  • Demonstrate that you have learned something.

  • Connect the topic to the real world.

  • Connect the topic clearly to the course, and involve the vocabulary and techniques that we have learned, as appropriate.

  • some math, don't just talk about it. Work a couple examples, prove a couple theorems, show me some diagrams.


Audience.
Your paper should be readable by your classmates. You should assume that your intended reader has had a basic course in linear algebra but is probably not expert in the specific application you have in mind. If you learn some new or more advanced linear algebra along the way, you should explain that. Take that When it's all over, I would like to make all these papers available to all of you, that we may all learn from one another. Let that serve as a final reminder of the huge variety of uses of linear algebra, if only you look for them.

Topics.
Chapter 10 contains twenty sections, each describing an application of the ideas of the course. Choose one and learn what it has to teach you. (Don't forget to look through the exercises and do a few … they can really help with understanding.) I can help you find additional materials to take your learning further if the text and Internet research feel unsatisfying. If you don't like any of these 20 topics, or if you have something else in mind, talk to me.

You're Not Alone.
This assignment is intended to be done by a pair of students working together for best results. You may choose to work individually, however. I will also allow groups of size three, but understand that my expectations will be higher from a group of
three. Students should register their partnerships (or individual status) with me as soon as convenient, but absolutely no later than April 20th. Changes to partnership arrangements after that must be approved.

Length.
Don't think too much about the actual page number length or word count. I'm not interested in filler or padding, I'm interested in how much you say. Here's a good rule of thumb. If you could give a short lecture (25-30 minutes) to your classmates on your topics, then you've learned enough. Tell me what you've learned.
Answered Same DayDec 21, 2021

Answer To: What is it? Students will write a short, formal expository paper about some application of the ideas...

David answered on Dec 21 2021
128 Votes
MARKOV PROCESSES OR CHAINS



As an application of Linear Algebra.




The prerequisites for the Markov Processes are :

Linear Systems

Matrices

Intuitive understanding of Limits
Markov Chains
A Markov Process
Let us consider a physical or that undergoes a process of change
such that at any moment, it can occupy
one of a finite number of states. Suppose that such a system changes with time from one state to
another and at scheduled times the state of the system is observed. Let us say, these changes are not
predictable, but they are governed by probability distribution. These changes incorporate a simple sort
of dependence structure, that is, the conditional distribution of future states of the system, given some
information about past states, depends only on the most recent piece of information. Which could be
simplified like, what matters in predicting the future of a system is its present state, and not the path by
which the system got to its present state.

If the state of the system at any observation cannot be predicted with certainty, but the probability that
a given state occurs can be predicted by just knowing the state of the system at the preceding
observation, then the process of change is called a Markov chain or Markov process.

Definition :

If a Markov Chain has k possible states, which we label as 1,2,...,k, then the probability that the
system is in state i at any observation after it was in state j at the preceding observation is denoted by
pij and is called transitional probability from state j to state i. The matrix P = [pij] is called transitional
matrix of Markov Chain.

In a three-state Markov Chain, the transition matrix is of the following form :


Preceding State

1 2 3


p11 p12 p13 1

p21 p22 p23 2 New State

p31 p32 p33 3



In this matrix, p31 is the probability that the system will change from state 1 to state 3 ; p22 is the
probability that the system will be state 2 if it was in state 2 ; and so on.




Let us look at two examples showing Transition Matrix of a Markov Chain

Example 1 :

A taxi service agency has three taxi stations, labeled by 1, 2 and 3. The taxi driver may pick up the car
from any of the one location and returns it to any of the three locations at the end of his working day.
The agency finds the taxi driver returns the car to the various stations according to the following
probabilities :

Picked from station

1 2 3

0.8 0.3 0.2 1

0.1 0.2 0.6 2 Returned to station

0.1 0.5 0.2 3



This matrix is the transition matrix of the system considered as Markov Chain.
We can infer from this matrix that, the probability that the driver picks the car from station 1 and
returns it to station 2 is 0.3. Similarly, the probability that he picks up the car from station 1 and returns
it to station 3 is 0.1.

Example 2 :

A survey was conducted by the blood donation society of a college. It...
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