Applications of Linear Algebra in the Real World

Choosing a Linear Algebra Topic and Real-World Example

In this course, we have used the weekly forum discussions to delve into Linear Algebra’s applications, asking such questions as: How is Linear Algebra used in the real world? Is it a common or uncommon technique? Is it used to cut costs? Increase computational power? Solve problems not otherwise easily (or cost effectively) solvable? Unfortunately, the weekly forum discussions by their nature have provided only limited opportunity to learn about how Linear Algebra is used in the real world. 

With that in mind, for this writing project we are going to delve in more depth into this issue. 

Please choose one Linear Algebra topic. You can choose any of the topics we’ve covered in this course, or even a topic that we did not cover. As long as it relates to Linear Algebra, it is fine. For a quick list of the topics we’ve covered, just look at the titles of the subsections of each chapter in the textbook.

Then associate your chosen topic with an example from the real world of how that topic can be applied to solve problems. Examples include (but are by no means limited to):

-Orthogonal Diagonalization in Statistics

-Linear Models in Civil Engineering

-Least Squares and GPS

There are numerous examples of Linear Algebra applications scattered throughout our textbook: in the first introductory pages of each chapter, in selected chapter sections (e.g., 1.10), and contained within the description of many homework problems located throughout the chapter. You can also find examples at any of the websites mentioned in the weekly forum descriptions, or at websites you research and find yourself.

Once you have picked a topic and application, you are ready to do research and write up your findings. For your chosen “Topic + Application” combination, be ready to answer the following questions:

1.Describe the Linear Algebra topic in general. Give a brief overview of the mathematics involved. If you are writing about (for example) orthogonal diagonalization in statistics, then your topic is orthogonal diagonalization and you would write an overview of it, much as you would find in a textbook. 

2.Describe the full range of applications for your chosen topic. Although the rest of your paper will be devoted to the single application of your chosen topic, for this section the idea is to paint a picture of the full range of applications that are possible. For example, orthogonal diagonalization can be used anywhere that you have large collections of data, and this occurs not only in statistics but also in image processing, weather forecasting, satellite surveillance, and the like. 

Be sure to list each of the different application areas and briefly (a sentence or two for each different application) indicate how your chosen topic is used for each type of application. For example, in weather modeling orthogonal diagonalization is used to model how the atmosphere will behave when the atmosphere is divided into small cells and each cell is treated as an individual data point

3.Now it is time to zero in on the specific “topic + application” combination that you have chosen. In this part of the paper, you should describe how the specific topic you chose is applied to real world problems. Essentially this is the same idea as item 2 above, but because you are focusing here on only your one single chosen application, you can go into it in much more depth and detail than you did for the brief overviews you wrote in item 2.

Continuing with our example, you might describe in detail how orthogonal diagonalization is used within statistics. What kinds of statistical applications is it used for? Could the statistical problems be solved any other way, and if so what is the advantage of using orthogonal diagonalization? To what extent, if any, is the process computerized and what changes, if any, are made to it in order to accommodate computerization? These particular questions probably do not apply directly to your own particular topic+application, but I am sure you can think of many similar aspects to discuss once you have done the research

4.The last part of the paper centers on doing some illustrative calculations. Create a numerical problem (the kind you might find in the textbook as a homework problem) – and then fully solve it. The problem you create should illustrate how the “topic + application” combination you chose is used in the real world. You may if you wish model your problem on one you find online or in the textbook, but you must use your own original numbers (not the numbers in the problem you modeled your version after) and then solve the problem yourself using your own numbers. Or you can write your own problem from scratch, in fact this is preferable if possible.