Teaching Linear Algebra: Technology and Resources

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Teaching Linear AlgebraTechnology and Resources

• Leslie Hogben
• Iowa State University, USA
• 3rd University Mathematics Courses Forum
• Chengdu, China
• November 2007

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Contents

• Technology
• Effect of LACSG
• Teaching linear algebra research
• Bibliography of linear algebra resources

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Technology

• Basic matrix computation
• Visualization
• Computation projects

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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

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Technology for basic matrix computations

• Calculators
• MATLAB, Octave
• Mathematica, Maple, SAGE

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Basic matrix computations

• determinant
• eigenvalues
• eigenvectors
• LU
• QR

• matrix arithmetic
• Inverse
• transpose
• RREF
• row operations

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Calculators TI 89, 92, Voyage 200

• Matrix operations
• Easy to use
• Hard to print

• Arithmetic
• Symbolic
• Exact
• Decimal

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Software MATLAB, Octave

• Matrix operations
• Easy to use
• Easy to print

• Arithmetic
• Symbolic ?
• Not exact
• Decimal

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Octave

• Free open-source download
• http//www.gnu.org/software/octave/octave.html
• Like MATLAB

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Software Mathematica, Maple, SAGE

• Matrix operations
• Hard to use
• Easy to print

• Arithmetic
• Symbolic
• Exact
• Decimal

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SAGE

• Free download
• http//www.sagemath.org
• Can also run on-line
• Excellent capabilities

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TI Calculators

• Entering a matrix
• 1,2,34,5,61,0,1 ? a
• Displays nicely

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TI Calculators Matrix operations
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TI Calculators Menus
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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

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file///Users/hogben/A20Leslie/A20research/07CHI
NA/HogbenUMCF/Linear20Algebra20Examples20(SAGE)
.webarchive
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Mathematica

• Entering a matrix
• a1,2,3,4,5,6,7,8,9
• Displays nicely only if told to

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Mathematica Matrix operations
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Mathematica Menus exist but are harder to use
than typing
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Visualization SoftwareRichard Vargas Gershini
http//etna.mcs.kent.edu/vol.8.1999/pp15-20.dir/ge
rshini.html
file///Users/hogben/A20Leslie/A20research/07CHI
NA/HogbenUCMF/gershini.webarchive
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Computation projects

• A project is
• A multi-part activity
• Involves technology
• Usually involves an application

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Sample Projects

• Markov chains
• Electric circuits (current/voltage only)
• Gershgorin circles
• Computer graphics

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Effect of LACSG

• Lead to ongoing discussion of teaching linear
  algebra
• Some criticism from math educators, but had very
  little effect
• Some recommendations widely adopted
• Others (omissions from core) ignored

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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.

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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

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What LACSG advice adopted?

• Matrix oriented
• Emphasis on Rn
• All core topics
• Applications
• Use of technology

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What was not adopted?

• Defer abstract vector spaces and linear
  transformations to 2nd course
• 2nd course at small colleges

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LACSG core

• Matrix oriented
• Emphasizes Rn
• LACSG core abstract vector spaces linear
  transformations
• Widely adopted

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LACSGs partitioned matrices
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Partitioned matrices in texts

• Partitioned matrix perspective partially evident
  in Lay, Leon, Strang,
• Fully in the graduate text by Zhang

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Iowa State University

• 21,000 undergraduates and 5,000 graduate students
• 140 undergraduate mathematics majors (many will
  become high school teachers)
• 50 PhD and 15-20 MS students in math and applied
  math

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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

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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

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Teaching Linear Algebra Research

• US-NSF Research Experiences for Undergraduates
  (REU) program
• ISU combinatorial matrix theory research group

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US-NSF REU

• US has a problem persuading undergraduates to
  enter graduate school
• REUs show students what doing math is like and
  create enthusiasm for math
• More REU students go to graduate school

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ISU Combinatorial Matrix Theory Research Group

• Summer REU for undergraduates
• Academic year early research course for first
  year graduate students

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Bibliography of Linear Algebra Resources

• Technology bibliography
• LACSG bibliography
• Undergraduate research bibliography

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Thank you!