## 2020-Malagon, Audrey - Remote Teaching Module - Geometry of Linear Systems

Mathematics, Virginia Wesleyan University, Norfolk VA USA

#### Abstract

Geometric Behavior of Linear Systems Module Overview

This module concentrates on the geometric properties of a system of 2 linear, homogeneous first order differential equations in 2 variables as dictated by the corresponding eigenvalues. Students should be familiar with using eigenvalues to solve linear, homogeneous first order systems before beginning this unit.

Prerequisites (Reviewed in the materials):

• Converting a 2 x 2 linear homogenous system to matrix form
• Finding eigenvalues and eigenvectors
• Creating general solutions using eigenvalues and eigenvectors (real and complex)

Module Components:

All videos are available by clicking on BoldFace Video and Number, but they are also available in Supportingt Docs for this General Resource for download.

• Video Series covering geometric properties of linear systems. This can be assigned to students in asynchronous or synchronous classes.
• Video 1: General review of how to convert 2 x 2 linear systems to matrix form and find eigenvalues
• Video 2: Geometric properties of systems with 2 distinct real eigenvalues
• Video 3: Geometric properties of systems with complex eigenvalues
• Video 4: Special cases (repeated and zero eigenvalues)
• Video 5: Trace Determinant Plane
• Slide Deck containing all slides in video series – can be used for synchronous presentation in remote format, or as a basis to create your own videos for students. (editable)
• Modeling Activity Student Handout Link (editable versions included)
• Modeling Activity Teacher Version with Solutions Link (editable versions included)
• Modeling Activity Video Series
• Video 1: Introduction to the scenario, guidance through creating a diagram and the system of differential equations.
• Video 2: Overview of the rest of the project and instructions on how to use free online graphing software.

https://homepages.bluffton.edu/~nesterd/apps/slopefields.html

https://www.desmos.com/calculator

• Modeling Activity Assessment Guide (editable)

Thanks to Kristin Burney, Virginia Wesleyan University, who provided the Mathematica images used in the presentation and gave valuable feedback on materials.

Audrey Malagon