# ModelCourses/Linear Algebra

## Contents |

# Model Courses/ Linear Algebra

## Introduction

This PREP Linear Algebra model course is a work in progress. Some of the problem sets below will likely be adjusted over the next several months (Winter/Spring 2012). Future updates will be indicated by the date when the changes are made.

A first-term Linear Algebra course could be taught in many different ways. A course in Linear Algebra could have more emphasis on applications than theory or vise-versa. The course could be taught on the quarter system or the semester system (or other time frame). Because of these and other variables, it is likely that an instructor who uses this model course will need to adjust some problem sets.

It is expected that an instructor will first import some or all of these problem sets into their WebWork course. Once the course is created and these problem sets are imported, there are a couple of options for how to use the problem sets. One way to use these sets is to make new problem sets and use the option in the WW library browser that allows one to search for problems from within the course (the local course created by the instructor). Essentially this method gives a refinement of the NPL, making it easier to find problems. Another option would be to use these problems sets and edit the sets as needed.

## Instructions for Importing and Exporting Problem Sets

Instructions for importing problem sets

Instructions for exporting problem sets

## Preliminary Topic List - 2011-06-23

- Vectors
- Geometric objects - lines and planes
- Dot product and Vector Projections
- Orthogonal decomposition

- Systems of equations and elimination
- Row operations and Row Echelon Form
- Gaussian elimination (Free variables & Consistency of solutions)

- Matrix operations and algebra
- Matrix arithmetic
- Matrix inverse
- Matrix equations
- Determinant and Cramer's Rule
- Elementary matrices and LU Decomposition

- Vector space preliminaries
- Definition of a vector space and subspaces
- Euclidean vector spaces
- Linear combinations and span
- Linear independence
- Basis and orthogonal basis
- Coordinate vectors and change of basis
- Row space, column space, and null space
- Dimension and rank
- Geometric examples

- Linear transformations
- Definition of a linear transformation
- Matrix of a linear transformation
- Reflections, rotations, dilations and projections
- Inverse of a transformation
- Kernel, range, injection, surjection

- Applications
- Graph theory: Adjacency matrix and Incidence Matrix
- Least squares
- Curve/surface fitting
- Mixture problems
- Simplex method
- Approximation of a function by a Fourier polynomial

- Eigenvalues and eigenvectors
- Finding eigenvalues and eigenvectors
- Eigenspaces
- Diagonalization
- Symmetric matrices & Trace
- Quadratic forms

- Inner product spaces and abstract vector spaces

## Problem Sets

### Unit 1 - Vectors

- Dot product and Vector projections, Orthogonal Decomposition
- Lines and Planes

Download 2 problem sets and header files as an archive. These files will be unarchived when uploaded to your WeBWork course:

Note: Rename this file without the .txt extension after downloading. Instructions for Windows users:

[7 or Vista] Start -> Control Panel -> Appearance & Personalization ->

Folder Options, click View, Advanced Settings, and unclick the "Hide extensions for known file types" check box.

[XP] My Computer -> Tools -> Folder Options -> View -> unclick "Hide

extensions..."

Chapter/sections used from the NPL that came from outside of Linear Algebra:

### Unit 2 - Systems of Equations and Elimination

- Row operations and Row Echelon Form
- Gaussian elimination
- Free variables & Consistency of solutions

Set definition file and set header file:

*(Both files should be renamed without the .txt extension after downloading)*