Matthew Eichhorn

MATH 2210

Over the summer of 2021, I worked to develop six new workshops and one new project for the department’s linear algebra course.

The workshops are completed in groups during a recitation section, and finalized and submitted by students individually over the following week. Each of the workshops is self-contained and allows students to explore an application of the topics that they are covering in lecture. Typically, six workshops are given throughout the semester.

The projects are completed individually and require students to carry out some research or data collection, calculation, and analysis. These projects are completed outside of class, similar to a homework assignment.

All of the section numbers correspond to Nicholson’s open access textbook Linear Algebra with Applications

Workshops

1: Balancing Chemical Reactions (Sec. 1.3)

MATH 2210 ∙ July 2021

Students set up a homogeneous linear system to balance each element in a chemical equation. This workshop demonstrates how understanding the solution space of a linear system can give insight into the feasibility of certain chemical reactions.

2: The Area of Polygons (Sec. 3.1)

MATH 2210 ∙ July 2021

Students make use of cofactor expansion in a geometric setting to derive the “Shoelace Formula” for the area of a general polygon given the coordinates of its vertices. Time permitting, students extend this formula to higher dimensions.

3: Hamming Codes (Sec. 5.4)

MATH 2210 ∙ July 2021

Students are introduced to finite field arithmetic. Then, they explore howto use use linear transformations to introduce redundency into a message vector, allowing for error detection and correction. Students get practice encoding and decoding using small Hamming codes.

4: Best Fit Curves (Sec. 5.6)

MATH 2210 ∙ July 2021

Students explore how we can not only use the method of least squares to approximate a linear function to data, but also to determine the best coefficients given any suitable basis functions. They are also introduced to the coefficient of determination, a metric for the quality of the approximation.

5: Group Representations (Sec. 6.2)

MATH 2210 ∙ July 2021

This workshop serves as a gentle introduction to the subject of abstract algebra. Students are introduced to the definition of a group and explore some matrix groups. Then, they study group representations, maps from arbitrary groups to matrices which “preserve” the group operation as matrix multiplication.

6: An Algebraic Integration Technique (Sec. 6.4)

MATH 2210 ∙ July 2021

Students recall the tricky “wrap-around” integration by parts calculations from calculus. By recognizing that the derivative is an invertible linear transformation for some choices of basis functions, we can solve these integrals using matrix multiplication.

Project

Curve Fitting (Sec. 5.6)

MATH 2210 ∙ July 2021

This project is a follow-up to Workshop 4. Students gather their own experimental data that models the quantitative relationship between two variables. They fit a linear model to the data, along with a custom-tailored model (chosen by plotting the data). Then, they use the coefficients of determination to analyze the quality of their models.