Teaching Linear Algebra: Technology and Resources

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Teaching Linear Algebra: Technology and Resources. Leslie Hogben Iowa State University, USA 3rd University Mathematics Courses Forum Chengdu, China November 2007. Contents. Technology Effect of LACSG Teaching linear algebra research Bibliography of linear algebra resources. Technology.

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Teaching Linear Algebra:Technology and Resources

Leslie Hogben

Iowa State University, USA

3rd University Mathematics Courses Forum

Chengdu, China

November 2007

Contents
• Technology
• Effect of LACSG
• Teaching linear algebra research
• Bibliography of linear algebra resources
Technology
• Basic matrix computation
• Visualization
• Computation projects
Use of technology for basic matrix computations
• Students learn and understand method (for example, RREF)
• Then use technology to solve:
• Larger more interesting problems
• Greater variety of problems
Technology for basic matrix computations
• Calculators
• MATLAB, Octave
• Mathematica, Maple, SAGE
matrix arithmetic

Inverse

transpose

RREF

row operations

determinant

eigenvalues

eigenvectors

LU

QR

Basic matrix computations
Matrix operations

Easy to use

Hard to print

Arithmetic:

Symbolic

Exact

Decimal

Calculators: TI 89, 92, Voyage 200
Matrix operations

Easy to use

Easy to print

Arithmetic:

Symbolic ?

Not exact

Decimal

Software: MATLAB, Octave
Octave
• http://www.gnu.org/software/octave/octave.html
• Like MATLAB
Matrix operations

Hard to use

Easy to print

Arithmetic:

Symbolic

Exact

Decimal

Software: Mathematica,Maple, SAGE
SAGE
• http://www.sagemath.org
• Can also run on-line
• Excellent capabilities
TI Calculators
• Entering a matrix:
• [1,2,3;4,5,6;1,0,1]  a
• Displays nicely:
SAGE
• Make matrix space M
• Then enter matrix as

A =M([[1,1/2,1], [3,-2,4/3], [4,-3/2,7/3]]); A

• Displays nicely

file:///Users/hogben/A%20Leslie/A%20research/07CHINA/HogbenUMCF/Linear%20Algebra%20Examples%20(SAGE).webarchivefile:///Users/hogben/A%20Leslie/A%20research/07CHINA/HogbenUMCF/Linear%20Algebra%20Examples%20(SAGE).webarchive

Mathematica
• Entering a matrix:
• a={{1,2,3},{4,5,6},{7,8,9}}
• Displays nicely only if told to:

http://etna.mcs.kent.edu/vol.8.1999/pp15-20.dir/gershini.htmlhttp://etna.mcs.kent.edu/vol.8.1999/pp15-20.dir/gershini.html

Visualization Software:Richard Varga’s Gershini

file:///Users/hogben/A%20Leslie/A%20research/07CHINA/HogbenUCMF/gershini.webarchive

Computation projects

A project is:

• A multi-part activity
• Involves technology
• Usually involves an application
Sample Projects
• Markov chains
• Electric circuits (current/voltage only)
• Gershgorin circles
• Computer graphics
Effect of LACSG
• Lead to ongoing discussion of teaching linear algebra
• Some criticism from math educators, but had very little effect
• Others (omissions from core) ignored
Why was LACSG influential?
• LACSG recommendations made sense to faculty teaching linear algebra
• Many universities and authors were already revising their linear algebra courses and texts in a similar manner.
Some LACSG recommendations were already happening
• In 1979 I taught linear algebra from a “LACSG-style” text (Anton)
• Matrix oriented, emphasized Rn
• Applications in a supplement
• Not much technology
• No use of partitioned matrices
• Matrix oriented
• Emphasis on Rn
• All core topics
• Applications
• Use of technology
• Defer abstract vector spaces and linear transformations to 2nd course
• 2nd course at small colleges
LACSG+ core
• Matrix oriented
• Emphasizes Rn
• LACSG core + abstract vector spaces + linear transformations
Partitioned matrices in texts
• Partitioned matrix perspective partially evident in Lay, Leon, Strang,
• Fully in the graduate text by Zhang
Iowa State University
• 140 undergraduate mathematics majors (many will become high school teachers)
• 50 PhD and 15-20 MS students in math and applied math
Linear Algebra at ISU
• In mid-1980s ISU faculty evaluated the first undergraduate linear algebra course (math 307)
• Course was split into two
• Both matrix oriented and use technology for basic computation
Linear Algebra at ISU
• Math 307 is a LACSG first course aimed at students in other fields
• Math 317 is a first linear algebra course for math majors and emphasizes proof writing but is matrix oriented and uses technology
• Math 471 is numerical linear algebra
Teaching Linear Algebra Research
• US-NSF Research Experiences for Undergraduates (REU) program
• ISU combinatorial matrix theory research group
US-NSF REU