Teaching Linear Algebra: Technology and Resources

1. Teaching Linear Algebra:Technology and Resources Leslie Hogben Iowa State University, USA 3rd University Mathematics Courses Forum Chengdu, China November 2007

2. Contents • Technology • Effect of LACSG • Teaching linear algebra research • Bibliography of linear algebra resources

3. Technology • Basic matrix computation • Visualization • Computation projects

4. 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

5. Technology for basic matrix computations • Calculators • MATLAB, Octave • Mathematica, Maple, SAGE

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

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

8. Matrix operations Easy to use Easy to print Arithmetic: Symbolic ? Not exact Decimal Software: MATLAB, Octave

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

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

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

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

13. TI Calculators: Matrix operations

14. TI Calculators: Menus

15. 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

16. 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

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

18. Mathematica : Matrix operations

19. Mathematica: Menus exist but are harder to use than typing

20. 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

21. Computation projects A project is: • A multi-part activity • Involves technology • Usually involves an application

22. Sample Projects • Markov chains • Electric circuits (current/voltage only) • Gershgorin circles • Computer graphics

23. 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

24. 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.

25. 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

26. What LACSG advice adopted? • Matrix oriented • Emphasis on Rn • All core topics • Applications • Use of technology

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

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

29. LACSG’s partitioned matrices

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

31. 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

32. 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

33. 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

34. Teaching Linear Algebra Research • US-NSF Research Experiences for Undergraduates (REU) program • ISU combinatorial matrix theory research group

35. 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

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

37. Bibliography of Linear Algebra Resources • Technology bibliography • LACSG bibliography • Undergraduate research bibliography

38. Thank you!