Lets Understand It With An Example
Assign the specific number to each alphabetical letter to encode the short secret messages.
Then, the sequence of the number of each text should be organized in the square matrix form .
Assume a nonsingular square matrix as B. To encode the message, multiply matrix A with matrix B . Assume the matrix B as:
and to encode the text, apply:
|Z = BA =|
Now, multiply the matrix Z to B inverse to decode the message:
As matrix multiplication is not commutative, it must maintain the matrices with the exact product value. If you multiply the Z and B inverse matrix in the opposite order, you will obtain:
Now, the secret message was CERNY PSIK .
Linear Programming In A Restaurant Setup
In a restaurant-based project, you can work with the following decision variables:
- Number of meals per day
- Different menu items
If the management wants to maximize the profit given the constraints on the menu like monetary budget and number of meals, we would require a simplification model. The number of food ingredients and the increase /decrease in meals are linearly related. Therefore, basic algebra applies to meal production in the restaurant kitchen, one of the most critical aspects of running a restaurant business. Cost and time would be some other limitations in such a problem.
You can use the above example to simulate a tangible eatery and create a plan for earning optimal profit. Alternatively, you can design a tool using the LP and graph approach to minimize the total kitchen preparation time for orders in a realistic setting.
Profit Optimization For Business Plan
Every business needs financial resources to get things started and sustain over time. Moreover, investors and credit providers usually ask for detailed analyses of profitability. And LPP can be immensely helpful in clearly explaining the viability of a business.
Let us suppose that you want to make a professional B-plan. First and foremost, you will have to define your problem using different variables, constraints, and the objective function. Then, you will solve the LPP using a suitable method, say a graph. Finally, you will evaluate your results and present your solution to the interested parties.
If you are a tutor, you can offer additional guidance and support by communicating the following details:
- Calculation worksheets
- Coding examples
Class projects on business plans enable immersive experiences as students get the opportunity to apply their imagination and come up with original business names and logos.
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What Is Linear Algebra
It is a branch of mathematics that includes vector spaces, infinite dimensions, and linear mapping among the spacesthe system of linear equations used for this kind of investigation.
The linear algebra equations are represented with the help of the formalism of vectors and matrices. Moreover, linear algebra is necessary for both applied and pure mathematics.
|Key Point:Linear algebra relates to linear combinations. It means you can use arithmetic on the numbers column that is known as arrays and vectors of the numbers termed as matrices. These matrices are created for new arrays and columns of the numbers.|
Lets take an example of it the abstract algebra method is used for relaxing the vector spaces axioms. The functional analysis use for the infinite-dimensional vector spaces version.
On the other hand, linear algebra helps in providing the linear systems solution for differential equations.
There are several techniques of linear algebra that are used for physics, computer science, analytic geometry, natural science, computer animation, and social science .
Being a well-developed theory, linear algebra is used for non-linear mathematics models approximated by linear details.
Why Is Linear Programming Important
Linear programming is a mathematical method to determine the optimal solution to a complex problem. As a problem can be thought of as a maximization of a linear combination of variables subject to a set of linear inequalities, it can be solved by linear programming. In linear programming, a combination of variables is used to maximize or minimize a linear objective function. Most problems seen in the real world have multiple objectives and constraints, so it is typical to use this method to find the optimum decision. Linear programming is used in economics, engineering and other fields. More precisely the area of application of linear programming includes transportation, location of facilities, finance, production and inventory control.
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Seeing The Forest: Linear Algebra Application Project
Educator: Ryan Bauer, Faculty and Assistant Department Head, MathematicsContext: Out of class Linear AlgebraKeywords: mathematics project, applied mathematicsStudent Activity Time: 6-8 hours
Students reflected on the course material in linear algebra and identified how the concepts and theories are applied in their field of interest.
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The semester just ended, and Im now in full retrospect mode. This semester I was fortunate to have only one prep two sections of Linear Algebra. Linear algebra, for me, is the cornerstone of a modern mathematics education precisely because its concepts and its mechanics lie at the heart of so much real-world stuff from web search algorithms to scheduling problems to computer graphics and many other areas. And yet, in a typical one-semester course on linear algebra you only get to touch on a handful of applications, and those tend to be sort of domesticated. A few years ago, I decided I wanted students to explore more than just the stock examples in the textbook, and I wanted them to do so in an authentic way that reflects real-world mathematical practice.
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Allocation Of Space In A University
A study conducted by researchers at Landmark University, Nigeria, applied the optimization principle to allocate classroom space. Based on the data obtained from lecture halls and the committees of examination timetable management, the project sought to maximize the usage of the available classroom space. The result was a framework that reduced overcrowding and utilized the existing seating capacity in the best possible way.
Casting Out Nines: Getting Students Involved With Linear Algebra Through Poster Projects
The semester just ended, and Im now in full retrospect mode. This semester I was fortunate to have only one prep two sections of Linear Algebra. Linear algebra, for me, is the cornerstone of a modern mathematics education precisely because its concepts and its mechanics lie at the heart of so much real-world stuff from
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Note On Extra Reading Material
For a different view on some of the topics presented in the textbook, you can look at sections from the following linear algebratext books:
- Introductory Linear Algebra with Applications, by Bernard Kolman
- Linear algebra, and introductory approach by Charles Curtis
- Computaional linear algebra with models, by Gareth Williams
NOTE: you are not expected to read all of the extra material avaliable,it’s just to give you more choice about what examples to use, and whatapplications to write about.
What Is Linear Programming
Linear programming is a simplification problem that seeks to replace a complicated mathematical expression. A typical case is based on three main factors.
Firstly, the simplification relationships should follow a straight line. Hence, the term linear. Secondly, all the values need to be subject to constraints, which may be in numerical or in terms or properties. And lastly, the solution has to optimize the quantity of a given variable.
Now that we have broken down LPP to its basic components, let us look at the terminology we use to formulate a problem.
- These variables determine the output and express the ultimate solution. For instance, the number of units of different cars manufactured by an automobile company .
- Objective function: It lays down what you want to achieve in a specific problem. For example, increasing profit, reducing lead time, etc. You would generally encounter the use of Z for the total profit function.
- Constraints: These are limitations or restrictions posed on the decision variables, such as the production capacity of machines, availability of inputs, etc.
- Non-negativity: As a rule, LPP decision variables take non-negative values, meaning that they are always greater than or equal to 0.
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What Are The Top Applications Of Linear Algebra
A spy detective game is one of my favorite games in my childhood. And I think you are not an exception who never play this game. In my childhood age, I always imagined how detectives decode secret messages? And I was always curious to know about it.
When I grow up, I am able to find out the method to decode the message. What is that? Well, it is one of the applications of linear algebra called Cryptography. Below, I have explained the exciting applications of linear algebra in detail. So, lets move to the details of one more concept of mathematics.
Linear Algebra In Economics
Write an essay on applications of linear algebra to problems ineconomics. Include either a description of the Leontief method, or something else. Write an essay on the applications of linear algebra tograph theory and network problems. Write how to use theseideas to solve either the problem of the graph theory game , or some other problem.
Other material avaliable:
- section 1-8 pages 73-97
- “Scheduling conflict-free Parties for a dating service”, Bryan L. Shader and Chanyoung Lee Shader, in the American mathematical Monthly, Feb 1997, Vol 104 #2.
Project On Supply Chain Management
This problem relates to warehousing and the transfer of manufactured products. Here, you would need to minimize the cost of transporting goods with respect to the resources available in the storage facility and at every depot.
You can take up a project to showcase the feasibility of linear programming in improving supply chain modeling. Consider an industrial case that determines the best transportation schedule within the given limits and establishes an integrated supply chain.
You can employ sensitivity analysis to demonstrate how small changes in the model data can change the optimal solution. Furthermore, your project can add to the existing body of research while highlighting the major issues in supply chain modeling.
Outline Of Suggested Form Of Essay
- Background description, overview:
- Not so much detailed mathematical content needed here, mainly motivational or historical,etc. The start of the chapters of the text book gives a good example.
- Mathematical ideas:
- This part should explain the mathematical content, explaining in words theconcepts, so that they can be easily understood.
- Computations, formulas, or examples:
- details of how these ideas are applied.
- diagrams, graphs or pictures
- Some visual component to the essay to convey the concepts in a clear way.
Note, you do not have to cover the above areas in the aboveorder, eg, you could start with some diagrams, and then explain theirmeaning, importance, what they are about.
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Linear Programming Projects On Automobile Production
Take the example of a company that has a product mix of four types of cars for example, luxury, sedan, minivan, and compact vehicle. The manufacturing facilities have a limit on the total annual production capacity. Additionally, the company has to comply with fuel consumption standards under various energy policies and national laws.
The details about the profit margins, market share, and fuel efficiency are available for all the models. Now, imagine a scenario where the top management wants to maximize the profits under these restrictions. It would be tricky to depict a strategy without simplifying the problem.
You can attempt a similar linear programming project based on the secondary data of any automaker. With this study, you can go a step further to analyze the implications of government policies on profitability. It is a unique way of looking at production planning problems that generally revolve around ERP data.
David Lay’s Projects And Case Studies
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Introducing The Reflection Activity
The transition from applied mathematics to more abstract mathematical concepts can be difficult for students, especially those who are taking their math courses to fulfill requirements for another major. One educator uses a project in a linear algebra course to prompt students to reflect on the course material and make meaning of the material in their discipline of interest. The purpose of this activity was to ignite students curiosity in mathematics, learn the relevance of linear algebra for their field of study, and to help them increase the value they associate with theoretical concepts in linear algebra.
Near the end of the term, the educator assigned a project centered around reflecting on the entire linear algebra course. Students were first prompted to find one concept, method, or theory taught in the course that applied to their major or field of study. The 2-3-page assignment required students to explain the context of their project and the concept from linear algebra, and provide a conclusion that included a reflection on what they learned from the project. The educator encouraged students to look beyond systems of equations and consider topics such as Eigenvectors or linear transformations that would be applicable to their field of study. The educator used a rubric to grade the assignment, which counted for 10% of the course grade.
Math 136 Project: Applications Of Linear Algebra
It may be that you’d like to choose something that’s not on thislist. If so, make an appointment to see me before Monday, May 2 so thatyou can get started on a more detailed outline of your proposedproject.
Each group should send me an e-mail message specifying your topic and thenames of everyone in your group. The deadline for doing this is 5:00pm,Thursday, May 5. To avoid unnecessary duplication, please send me onemessage per group .
Acknowledgement.I obtained the material below from RebekahHahn, one of our former graduate students.
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Previous Projects From 2270 Courses
Projects from 2270 in 2012, Strang’s TextbookHere.
Presented Projects 2012 Directory Link 2012 Aldous, Arnold and Edwards: Waveforms and Spectrograms Down, Firestone and Reed: Fractals: Iterated Function Systems and Linear Algebra Edfrennes, Roddum and Thorsen: Cracking the Code: An Introduction to Hill Ciphers Guckert: Fast Fourier Transform and the Modified Discrete Cosine Transform in MP3 Audio Compression Kubly and Pellatt: Forecasting United States Real Gross Domestic Product McGrath: Using Linear Algebra to Determine Spatial Autocorrelation: Geography Weeks: The Vertex Adjacency Matrix: Illustrated Tales of The Tortoise The Spanning Tree The Eel Yizhou Ye: Image Compression by SVD and DCT Submitted Projects and Incomplete Drafts 2012 Directory Link 2012 Azad and Wiser: Image Editing: Photos, RGB and Linear Algebra Bess: Image Compression via DCT and SVD: A Matlab Investigation Boyer: Gaussian Quadrature: An Application of Gram-Schmidt Christensen: A Brief History of Linear Algebra Gautam: Markov Chains and Nepal Voting Behavior Koizumi: Sound Compression of WAV Files: Maple Investigation Boya Li: Productive Economy: A Maple Investigation Partridge: Fractals: A Maple Investigation Wang and T. Ye: Relationship between Economic GDP and Mathematics
Projects from 2270 in 2016, Lay’s TextbookProjects from 2270 in 2017, David Lay’s Textbook
Exploring Linear Algebra: Labs And Projects With Mathematica
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What Are The Limitations Of Linear Programming
Linear programming is a powerful tool however, it can’t do everything. It is only as good as the data that is input. If the data is bad, the results will be bad. The best way to get good results of linear programming is to make sure that your data is set up properly and is accurate. Also, linear programming can only handle ‘solved’ problems. You can’t use linear programming to determine the best course of action for a problem that hasn’t been solved yet. For example, linear programming can’t help you find the best way to get from your house to your job. It can help you determine the best way to get to your job after you have figured out how to get to your house.