Constructing a scientific report is fundamental in academia and industry, especially in the context of applied linear algebra. Your expository project provides an opportunity for you to engage in the challenges of writing a paper that revolves around exploring and understanding the theory of linear algebra. This process enables you to apply key linear algebraic principles to analyze and interpret data (real or simulated), contributing to a deeper understanding of the subject matter.
The most important aspect of your project is that you must write a reproducible report. That means that you must include code into your report - similar to some homework assignments. This can be done easily using RMarkdown for R or Jupyter Notebook for Python. Your submission should be a pdf or html file.
Since some parts of the project is computational, you will be required to use a computational tool of your choosing. You can choose either R or Python. These tools are free and open-source. See the Syllabus for Cloud Computing.
| Topic | Description |
|---|---|
| Population Dynamics | Applied linear algebra plays a crucial role in modeling and analyzing population dynamics. Linear algebraic methods, such as matrix operations and eigenvector analysis, help in understanding the interactions between different demographic factors, predicting population trends, and studying the impact of various factors on population growth or decline. These mathematical tools provide a quantitative framework for studying birth rates, death rates, migration patterns, and other demographic phenomena. |
| Game Theory | In the realm of game theory, applied linear algebra is employed to model and analyze strategic interactions between rational decision-makers. Matrices and vectors are used to represent the payoffs and strategies of players in various scenarios. The concept of Nash equilibrium, a central idea in game theory, often involves solving systems of linear equations. Linear algebraic methods provide a powerful foundation for studying competitive and cooperative behaviors in diverse strategic situations. |
| Image Processing | Applied linear algebra is fundamental to image processing, where images are represented as matrices of pixel values. Operations such as convolution, filtering, and transformations are efficiently implemented using linear algebraic techniques. Matrix manipulation allows for tasks like edge detection, image enhancement, and feature extraction. Singular Value Decomposition (SVD) is often applied for image compression and denoising. Linear algebra enables the development of algorithms that enhance and analyze digital images for various applications. |
| Natural Language Processing | In the domain of Natural Language Processing (NLP), linear algebra plays a crucial role in representing and processing textual data. Word embeddings, document representations, and semantic analysis are achieved using linear algebraic structures such as matrices and vectors. Techniques like Singular Value Decomposition (SVD) are employed for dimensionality reduction, enabling more efficient processing of large textual datasets. Linear algebra empowers NLP models to capture and understand semantic relationships between words and documents. |
| Social Networks | Applied linear algebra is instrumental in modeling and analyzing social networks. Networks can be represented as adjacency matrices, and linear algebra provides tools to analyze connectivity, identify influential nodes, and study the overall structure of social graphs. Eigenvector centrality is often employed to measure the importance of nodes in a network. Linear algebraic methods enable researchers to uncover patterns of influence, study information flow, and analyze the dynamics of social interactions within a network. |
| Bioinformatics | In the field of bioinformatics, applied linear algebra is used for various tasks such as analyzing genetic data, sequence alignment, and molecular structure analysis. Linear algebraic methods help in representing biological data as matrices and solving computational problems efficiently. Techniques like Principal Component Analysis (PCA) are applied for dimensionality reduction in high-dimensional biological datasets. Linear algebra facilitates the extraction of meaningful information from biological data, aiding researchers in understanding complex biological systems. |