Applications: Projects requiring solutions of systems

Solving systems of equations

Nodal analysis of circuits - Uses systems of equations to find the current through each loop of a circuit including batteries and resisters.  Nodal analysis creates linear equations using Kirchhoff's Laws of junctions and paths.  This is a popular project for students who have studied some physics.  One value of this project is the ability to create overdetermined, consistent systems of equations, which helps students understand rows of zeros in the RREF form of augmented matrices.  This article on Nodal Analysis of Electric Circuits has a clear explanation.

Loop analysis of circuits - Uses systems of equations to find the current through each loop of a circuit including batteries and resisters.  Loop analysis creates linear equations using Kirchhoff's Laws of loops.  Again, a popular project for students who have studied some physics and also has the opportunity for overdetermined, consistent systems.  Equivalent in results to nodal analysis, this could be combined or assigned separately.  This article on Loop Analysis of Electric Circuits has a clear explanation.

Curve fitting - Using systems of equations a student finds the coefficients of a polynomial of degree n - 1 to fit n points.  I don't think of this as a juicy application that gives the student an appreciation for how linear algebra is used in the world.  Fitting an n-degree polynomial to m points using least squares or other methods is more likely to happen.

Applications: Projects using vector spaces

Inner product spaces

Curve fitting using least squares - Uses matrix multiplication, inverses, and equations to find coefficients of a curve that fits a set of points.  I have three misgivings about curve fitting with least squares: it can be done without any understanding, it is not representative of least squares problems in general, and the various spaces involved confuses the issue.  Taking the last point first, the problem involves points in 2D or 3D space, matrices in n x k space where n is the number of points and k the terms of the curve being fit, and coefficients that live in k-space.  If we are trying to fit a line, the coefficients are 2D, but that 2D space is not the 2D space of the original points.

If a student is assigned this project without have learned about projections, they can do the calculations anyway, since they just require matrix multiplication and solving systems of equations.  The process of setting up the matrices does not promote deeper understanding of inner product spaces, and so if the student is going to fit curves, they might as well use Excel, which also doesn't deepen their understanding of linear algebra.

Finally, if a student learns least squares in this way, then they have difficulty transferring this concept to the solutions of noisy systems using least squares and don't think of least squares as a method of approximating solutions, but rather of fitting curves.

Applications: A list of projects using matrix operations

Matrix Operations

I've just finished teaching Linear Algebra twice since the beginning of the year, and I'll be teaching it again in the fall. It is time that I cleaned up my applications list and updated the project files. Since I am enumerating them, I might as well do it here. Here is part 1 on matrix operations.  Others may be added later.

Seriation in archaeology - Uses incident matrices, matrix multiplication and transposition, and the properties of symmetric matrices to determine relative ages of sites with common artifacts.  I haven't used this in my classes yet, but there seem to be some good resources available including "Some problems and method in statistical archaeology," David Kendall, World Archaeology, 1969, which is available in JStor.  It also appears in Gareth Williams Linear Algebra with Applications.

Color manipulation in images - Uses matrix multiplication to alter colors in the RGB scale.  This article by Paul Haeberli describes the 4x4 matrices needed to modify colors, including offsets.  I haven't used this in class yet, but I could see this as a good project to have students work in Mathematica.  It also is a companion for transformations in 3D graphics, which also use 4x4 matrices.  The ability to use matrix multiplication to add vectors is common to both areas.

Image color conversion - Uses matrix multiplication to convert from, say, RGB to YIQ, color models.  I haven't used this in class, but it shows up in Gareth Williams Linear Algebra with Applications. This article by Ford and Roberts describes a bevy of color models, and it seems that only some conversions are linear.  Without a way to test whether the color conversion is correct, I don't see this as an interesting project. However, maybe Mathematica can render the other color models.

Transformations in 2D graphics - Uses matrix multiplication to apply rigid and non-rigid transformations to images.  May or may not use projective coordinates, depending on whether translations are allowed.  Resources abound.

Projection of 3D images onto the plane - Uses matrix multiplication to project 3D images onto the plane given the coordinates of the image and the location of the viewer.  Uses projective coordinates.  I use a paper written by Jeanie Mullen, one of my honors students.  This project has worked best for students with programming backgrounds.

Two-port in an electrical circuit - Uses matrix multiplication to describe the change in voltage and current through a two-port or a series of two-ports.  A simple application of Ohm's law that creates two linear equations that can be described using matrix multiplication.  The equations relate the input current and voltage to the output current and voltage.  This Wikipedia article has a table of many transmission matrices and their effect.

Applications: A list of projects using eigenthings

Eigenthings

Gould's accessibility index in a network - The process uses a modified adjacency matrix and the components of the eigenvector associated with the dominant eigenvalue. Students find this approachable and adaptable.  Applications to historical geography, air traffic.

Discrete dynamical systems - Using linear algebra to study discrete dynamical systems comes in several flavors.  Here are some projects that students find interesting and that differ from each other enough that they feel they are not repeating someone else's project.

• Difference equations and the Fibonacci sequence - Using eigenvalues to write the product of the nth power of a diagonalizable matrix and an initial vector allows one to write a closed form for a recursive formula.  Matrices of size 2x2 are needed to write the closed form of the nth Fibonacci number, but students can easily move from there to the closed form of 3rd and 4th order difference equations.  This project is always chosen by some student even though it is not applied to a real-world situation.
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Applications: A list of projects using matrix inverses

Matrix Inverses

I separate these projects from those other using matrix operations, because I make a clear distinction between forward and inverse problems in my classes.

Cryptography - These projects come in two varieties: using modular arithmetic and not.

• Matrices with |determinant| = 1 - Uses matrix multiplication to encode a matrix and multiplication of the inverse to decode.  Any matrix with determinant 1 or -1 will result in an inverse with integer components.  Students tend to be drawn to these projects, but sometimes I find it hard to push them further, such as requiring them to create their own encoding matrices, etc.  Resources abound.
• Modular arithmetic and row reduction - Uses matrix multiplication in modular arithmetic to encode and decode a message and row reduction in modular arithmetic to find the inverse.  This is not that the decoded matrix is read using mod 26, but rather that the matrix operations are done with, say, mod 37.  This project requires a little more tenacity on the part of the student, and this article by Keith Conrad discusses inverses of matrices under modular arithmetic.