TECHNOLOGY IN INTRODUCTORY LINEAR ALGEBRA: PROJECTS AND PEDAGOGYJOINT MATHEMATICS MEETING

SEATTLE, WA

2016 PAT GARDNER

CONTEXT

• Columbia Basin College: Linear Algebra students are aspiring

“general practitioner” engineers

• Author: Adjunct Mathematics Instructor with long history and

interest in applications of technology to teaching and learning

AGENDA

• WAMAP -- online teaching and testing support tool

• Course Projects (Mathematica)

• Grand Project – Signal Processing

WAMAP – SVD

WAMAP – SPECTRAL DECOMPOSITION

WAMAP -- EIGENVALUES 1

WAMAP -- EIGENVALUES 2

EXERCISE

Write a (simple) program to randomly generate a 2x2 matrix that

a. has exactly one eigenvalue, and

b. whose eigenspace has dimension 1

COURSE PROJECT

Reference. Anton, Rorres, Elementary Linear Algebra, 10th ed., section 10.1

PROJECT TEMPLATE

FAVORITE LINEAR ALGEBRA PROBLEM

GRAND PROJECT

Name That Tune

Linear

GRAND PROJECT – SIGNAL PROCESSING

Inspiration: MIT Open Course 6.003

Technologies:

• SciLab

• MAKE: Electronics Kit

MIT OPEN COURSE 6.003

RLC CIRCUIT

Ref. https://en.wikipedia.org/wiki/RLC_circuit

MODELICA (SCILAB)

RESULT

LINEAR ALGEBRA

Ref. http://www.openeering.com/sites/default/files/Control%20System%20Toolbox%20in%20Scilab.pdf

STATE SPACE REPRESENTATION

GOAL: CONSTRUCT AND ANALYZE

Ref. https://www.youtube.com/watch?v=rTmlR8B-u5c

EXAMPLE. SIMPLE SYNTHESIZER

Topics• V = IR• 𝑃 = 𝐼2𝑅• Graphs, functions,

(Dis)continuities• Time constantExponential function

𝑉𝐶 =1

𝐶 𝐼

EXAMPLE. SIMPLE SYNTHESIZER

Requested: 51 Ohm resistor

Have: • 100 Ohm resistors (10)• 180 Ohm resistors (10)• 330 Ohm resistors (10)• 470 Ohm resistors (10)

Exercise: Now what?

EXAMPLE. SIMPLE SYNTHESIZER

Requested: 51 Ohm resistor

Have: • 100 Ohm resistors (10)

1

𝑅=

1

𝑅1+

1

𝑅2

EIGENVALUES

•Hear λ?

• Pure tone

• Square wave – computer vs. mechanical

• See λ?

• Connect an LED in an RLC circuit?

• Time constant = L/R: 𝑒− (𝑅 𝐿)𝑡 factor

EXAMPLE. RLC CIRCUIT STATE SPACE

• State vector 𝑖(𝑡)

𝑑𝑖 𝑑𝑡

• battery source

Initial condition?

Yeah, that's a serious question. Maybe, can I hang on until Friday, and even maybe next week. Because it's very important. When I said boundary conditions are the key to problems, I'm serious. If I had to think okay, what do people come in my office ask about questions, I say right away, What's the boundary condition? Because I know that's where the problem is.

EXAMPLE. RLC CIRCUIT STATE SPACE

• State Variable: 𝑥 =𝑉𝐶𝐼

•𝑉𝐶𝐼

𝑑𝑜𝑡

=0 1 𝐶

−1𝐿 −𝑅

𝐿

𝑉𝑐𝐼

+0 1 𝐿

𝑉𝑖𝑛

• Initial Condition: 𝑥 0 + =𝑉𝐶(0+)𝐼(0+)

=00

GRAND PROJECT DESIGN GOALS

•Connect mathematics to the physical world.

•Work same example problem(s) in multiple courses at appropriate level of expertise

FOCUS, DEC 2015/JAN 2016, P. 26

ACKNOWLEDGEMENTS

• Gil Strang, Linear Algebra And Its Applications

• David Arnold, College of the Redwoods, student linear algebra projects (http://msemac.redwoods.edu/~darnold/math45/laproj/)

• MIT Opencourseware, esp. 18.06, 18.085, 6.003

• Charles Platt, Make: Electronics

• Michael Ciletti, Circuit Analysis and Design

• Charles MacCluer, A Survey of Industrial Mathematics

• Walter Rudin, Functional Analysis

RESOURCES

• Available Materials

• These slides

• projectTemplate.nb, .cdf -- Mathematica source (nb) and executable (cdf) file for constructing curves and surfaces passing through given points; cf. Anton/Rorres, Introductory Linear Algebra, 10 ed., section 10.1.

• SciLab files (sce and zcos) implementing multiple representations an RLC parallel and an RLC series circuit based on openeering.com’s “Introduction to Control Systems in SciLab;” includes additional documentation, rlcExamples.docx

• Contact: pgardner@columbiabasin.edu