Tuesday, 14 February 2012

Linear independence vs determinants

When writing my blog post on threshold concepts I read some interesting comments from Gordon Royle on linear independence versus calculating determinants:
In first year linear algebra (which is the current cross I have to bear), the concept of “linear independence” seems to meet these requirements. Many students struggle for weeks with the definition, but then suddenly (for the lucky ones) “the penny drops” and it all seems totally obvious. Once they have mastered linear independence, suddenly finding bases and calculating dimensions becomes easy and natural. On the other hand, calculating the determinant of a matrix is also something that students struggle with, but it is not a threshold concept. It’s just memorizing the rules and the signs that some students find difficult. Mastering it doesn’t really lead them anywhere except for being able to calculate the determinant of a matrix.
I agree that the mechanical operation of calculating the determinant of a matrix is not a threshold concept. Around 1990 Bruno Buchberger (RISC, Linz) wrote:
Many areas of high school and undergraduate mathematics, by now, are "trivialized" in the sense that their problems can be solved algorithmically by existing mathematical software like Mathematica. Well, if an area of mathematics is trivialized, why should students bother to study the area? Rather, shouldn't we just teach the students how to solve the main problems in the area by applying, in a reasonable way, the appropriate algorithms in, say, Mathematica? There are two dogmatic answers to this question. The puristic answer: Ban math software systems from math education! The pragmatic answer: Don't spend time in class on any trivialized area of mathematics!
Buchberger developed the white-box/black-box principle for math education using math software, which I subscribe to:
The 'white-box' phase is traditional teaching (study of underlying theory together with practice examples). Once a topic is understood, software may be employed as a 'black-box' , and so on recursively as the student progresses through a mathematics course.
So I would argue that little time should be spent on hand calculating determinants. However, I take issue with Gordon's statement that
The trouble with our existing teaching materials (for my first year unit) is that the time and number of pages spent on defining linear independence and calculating determinants is almost exactly the same. Each of them are simply defined and a couple of examples given. No student would have a clue (unless they were told) that one is a peripheral computational technique and the other is a critical conceptual issue.
Understanding what a determinant is, its properties, and how it can be used (applications) is, I think, definitely a threshold concept, not a peripheral computational technique. Examples (from my MATH2200 course):
  • If two rows (or columns) of a matrix are the same (or proportional), or a linear combination of two yields another row then, without any further computation, the determinant vanishes (an example of linear dependence);
  • Swapping a pair of rows or columns changes sign (no computation required); 
  • Learning the Laplace (or cofactor) expansion, along with the above observations, is useful: orthogonal polynomials can be defined as the determinant of simple tridiagonal matrices and properties such as their recurrence relation are easy to deduce from this;
  • The determinant, like the trace, is a matrix invariant equal to the product of the matrix eigenvalues. This is a useful check on the computation of eigenvalues;
  • The matrix invariants of A are generated as powers of x in the expansion of the characteristic equation, |A - x I|;
  • Computation of Wronskians;
  • Determination of equations of conical sections passing through a set of points;
  • Wavefunctions for Fermions in quantum mechanics are antisymmetric under particle interchange and are constructed using the Slater determinant;
  • Using the Levi-Civita symbol and Einstein summation notation is a "simple" way to view the determinant.

1 comment:

  1. I guess I should clarify that my comments were only about "the hand calculation of determinants". The only concept associated with determinants in those (now superceded) notes to which I was referring was "invertible if and only if non-zero determinant" which doesn't appear to cause any problems to students... at least superficially.