Final Project (Proposal)
Describe a particular problem from another discipline that can be addressed through the use of one or more of the linear algebra techniques was have studied this semester.
TOPIC IDEAS (You should not re-use a project done in any other class.) Note that you must apply in some way a topic that has been introduced in the class post-exam 2: coordinate systems, eigenvalues, diagonalization, etc.
Markov Chains – Model a board game, sport, or real-world situation using a Markov chain, then determine optimal strategies through an analysis of the model. Or use the basic ideas behind Google’s PageRank algorithm for ranking the popularity of Web pages to rank some other set of connected items. If you choose this topic you may need to model a very simplified version due to computational constraints. These can also be applied to biology, for example, model inheritance of genetic traits
Discrete Dynamical Systems – Analyze a population, perhaps using actual birth, survival, and death rates, using the eigenvalue approach to dynamical systems (this is covered later in our textbook in Chapter 5—you can read ahead if you are interested in this). Use the Leslie model to solve a problem in ecology; e.g. look at Hawaii’s population and analyze it with actual birth, survival and death rates
Social networks analysis using graphs and adjacency matrices.
Obtaining a closed formula for the Fibonacci Sequence. (This and other “pure math” applications are acceptable.)
Explain the proof or application of the Jordan Canonical Form of a matrix. This is a deep theorem of interest to those in pure mathematics, as well as a very useful formula that appears in a variety of applied mathematics scenarios.
Choose one of the applied mathematics projects from the Boyce Applied Mathematics textbook. (Knowledge of differential equations required!) Or look at 5.7 in our book about applications of differential equations.
Use real-life season data to create a win-loss differential matrix and obtain the rankings of sports teams.
Section 5.6 : Discrete Dynamical Systems and predatory-prey models
Learn how to tell when a linear transformation is 1-1 and onto. Show that any finite dimensional vector space is isomorphic to ℝ n using the coordinate map.
Section 5.4: Eigenvectors and Linear Transformations.
Section 7.4: The Singular Value Decomposition and applications to machine learning.
Finding the nth power or a square matrix; finding the exponential of a matrix