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

Technology  Basic matrix computation  Visualization  Computation projects  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  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  M ATLAB, Octave  Mathematica, Maple, SAGE  Calculators  M ATLAB, Octave  Mathematica, Maple, SAGE

Basic matrix computations  matrix arithmetic  Inverse  transpose  RREF  row operations  matrix arithmetic  Inverse  transpose  RREF  row operations  determinant  eigenvalues  eigenvectors  LU  QR

Calculators: TI 89, 92, Voyage 200  Matrix operations  Easy to use  Hard to print  Matrix operations  Easy to use  Hard to print  Arithmetic:  Symbolic  Exact  Decimal

Software: M ATLAB, Octave  Matrix operations  Easy to use  Easy to print  Matrix operations  Easy to use  Easy to print  Arithmetic:  Symbolic ?  Not exact  Decimal

Octave  Free open-source download  octave/octave.html octave/octave.html  Like M ATLAB  Free open-source download  octave/octave.html octave/octave.html  Like M ATLAB

Software: Mathematica, Maple, SAGE  Matrix operations  Hard to use  Easy to print  Matrix operations  Hard to use  Easy to print  Arithmetic:  Symbolic  Exact  Decimal

SAGE  Free download   Can also run on-line  Excellent capabilities  Free download   Can also run on-line  Excellent capabilities

TI Calculators  Entering a matrix:  [1,2,3;4,5,6;1,0,1]  a  Displays nicely:  Entering a matrix:  [1,2,3;4,5,6;1,0,1]  a  Displays nicely:

TI Calculators: Matrix operations

TI Calculators: Menus

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  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).webarchive

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

Mathematica : Matrix operations

Mathematica: Menus exist but are harder to use than typing

Visualization Software: Richard Varga’s Gershini 20.dir/gershini.html file:///Users/hogben/A%20Leslie/A%20research/07CHINA/Hogben UCMF/gershini.webarchive

Computation projects A project is:  A multi-part activity  Involves technology  Usually involves an application 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  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  Some recommendations widely adopted  Others (omissions from core) ignored  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

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.  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 R n  Applications in a supplement  Not much technology  No use of partitioned matrices  In 1979 I taught linear algebra from a “LACSG-style” text (Anton)  Matrix oriented, emphasized R n  Applications in a supplement  Not much technology  No use of partitioned matrices

What LACSG advice adopted?  Matrix oriented  Emphasis on R n  All core topics  Applications  Use of technology  Matrix oriented  Emphasis on R n  All core topics  Applications  Use of technology

What was not adopted?  Defer abstract vector spaces and linear transformations to 2nd course  2nd course at small colleges  Defer abstract vector spaces and linear transformations to 2nd course  2nd course at small colleges

LACSG+ core  Matrix oriented  Emphasizes R n  LACSG core + abstract vector spaces + linear transformations  Widely adopted  Matrix oriented  Emphasizes R n  LACSG core + abstract vector spaces + linear transformations  Widely adopted

LACSG’s partitioned matrices

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

Iowa State University  21,000 undergraduates and 5,000 graduate students  140 undergraduate mathematics majors (many will become high school teachers)  50 PhD and MS students in math and applied math  21,000 undergraduates and 5,000 graduate students  140 undergraduate mathematics majors (many will become high school teachers)  50 PhD and 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  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  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 Research Experiences for Undergraduates (REU) program  ISU combinatorial matrix theory research group

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  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

ISU Combinatorial Matrix Theory Research Group  Summer REU for undergraduates  Academic year early research course for first year graduate students  Summer REU for undergraduates  Academic year early research course for first year graduate students

Bibliography of Linear Algebra Resources  Technology bibliography  LACSG bibliography  Undergraduate research bibliography  Technology bibliography  LACSG bibliography  Undergraduate research bibliography

Thank you!