technology in introductory linear algebra: projects and ... · boundary conditions ... implementing...
Post on 19-Apr-2018
223 Views
Preview:
TRANSCRIPT
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
EXERCISE
Write a (simple) program to randomly generate a 2x2 matrix that
a. has exactly one eigenvalue, and
b. whose eigenspace has dimension 1
GRAND PROJECT – SIGNAL PROCESSING
Inspiration: MIT Open Course 6.003
Technologies:
• SciLab
• MAKE: Electronics Kit
LINEAR ALGEBRA
Ref. http://www.openeering.com/sites/default/files/Control%20System%20Toolbox%20in%20Scilab.pdf
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
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
top related