Analysing subspaces, linear independence and eigenvectors

I decided to analyse three concepts: subspaces, linear independence and eigenvectors/ eigenvalues. In deciding which concepts to focus on I referred to my analysis of the research meetings and to the research literature into the teaching and learning of linear algebra. I was guided by the lecturer’s own view, expressed in the research meetings, as to what the ‘crucial concepts’ [his words] of linear algebra were. The lecturer said,

That chapter 3 on subspaces. ... It’s the heart of the course really, that’s why I told students on Friday that if they master that chapter everything else will fall into place easily. And I’m convinced it will. And, on the other hand, that chapter is, well, where all the abstract concepts really are. That’s the reason why it’s so difficult. (M18, 27:42)

Chapter 3 (of the course notes) was entitled “Subspaces ofRn”. It was one of four chap- ters that constituted the course notes for the linear algebra module. In terms of linear algebra concepts, in Chapter 3, the lecturer introduced the concepts of vector, linear transformations/linear maps, null space, subspace, linear combination, span, spanning set, range of a matrix, linear (in)dependence, linear relation, basis, dimension, rank, nullity, rank-nullity theorem, and change of basis to students. In the exposition of these concepts in the course notes the lecturer did not give an indication of one concept being more important than another. However, in research meetings and during his teaching he often stressed the importance of certain concepts. For example, in a lecture in Week 5 of the module the lecturer told students explicitly that a subspace was an important concept. He said,

And because this is so important we give this a name. We call this a subspace of Rn. A subspace of Rn is a subset that behaves the same way as the full space Rn in the sense that I can add two vectors and get another vector in the set. I can multiply a vector from the set with a number and get another vector in the set. And, the zero vector is in the set. (L12, 40:53)

A little later in the lecture he said,

That’s [A subspace is] an important new idea and also an important new word that you need to learn to use. And there’re a couple of more new words that I need to give you. (L12, 43:25)

Hence I decided to analyse the concept of a subspace in more detail.

A second concept that the lecturer considered ‘crucial’ [his words] was linear indepen- dence. In a research meeting at the beginning of Semester 1 the lecturer talked about the changes that he had made to the module. He said,

Well, I am making quite some changes to the material that I am covering in lectures as compared to last year. Because, I am hoping to put more emphasis

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on the crucial concepts of linear algebra, that are there when you look at the module as a whole which is linear combinations, linear independence, bases. (M1, 41:50)

In a lecture in Week 6 the lecturer introduced and discussed the concept of linear inde- pendence. In defining this concept he said to his students,

. . . , when we’re given a set of vectors, a spanning set, we can investigate if there is a linear relation between the vectors. If there is, we say the vectors are linearly dependent, if there is none, we say the vectors are linearly independent. . . . That is quite possibly the most important idea you’re going to see in this chapter, and in all of the module, because something very similar crops up all over the place in mathematics. (L15, 24:42)

In addition I consulted the research literature into students’ difficulties with linear alge- bra. In particular, I considered the work by Jean-Luc Dorier and his colleagues (Dorier, 2000; Dorier, Robert, Robinet and Rogalski, 2000b). Concepts mentioned specifically by Dorier were vectors, basis and dimension, linear independence, and rank.

My review of the research literature into the teaching and learning of linear algebra (see Chapter 2) supports the view that linear independence is an important concept in linear algebra.

Hence I decided to analyse the concept of linear independence in more detail.

The third concept that I decided to analyse was eigenvectors and eigenvalues. “Eigen- values and Eigenvectors” was the title of Chapter 4 of the course notes. In research meetings the lecturer talked extensively about structuring the material of Chapter 4 (and the introduction to eigenvalues/vectors, in particular) in order to present students with a more conceptual view of this topic. He said,

. . . trying to do the important concepts first and the computational recipes later on, in order to avoid that students think of an eigenvalue as a zero of the characteristic polynomial, which most know anyway because that’s how they’re calculated. But this point of view really is unhelpful if you want to move out of the calculation because it doesn’t give you an opportunity to do anything with it. (M15, 1:10:26)

Eigenvalues/vectors are a pre-requisite for dealing with change of bases/diagonalisation of matrices (also in Chapter 4 of the course notes). They also provided the focus for re- cent research into the teaching and learning of linear algebra (see Stewart, 2009; Stewart and Thomas, 2007, 2010).

I have chosen three topics, subspaces (and vector spaces in general), linear independence and eigenvalues/vectors, for further analysis. In a research meeting the lecturer listed these three topics among several that he considered as making up a ‘standard’ university module in linear algebra. The lecturer said,

. . . we certainly all agree we want students to be able to do matrix calcula- tions and solve linear equation systems and to calculate eigenvectors, and we also want them to understand the concepts of vector spaces and linear independence, and so, I think we all agree on that. And to that extent In- troductory Linear Algebra really is standard, not only here but everywhere. (M18, 06:30)

Guided by the lecturer’s comments in research meetings and by the research literature into the teaching and learning of linear algebra I decided to analyse the concepts of subspaces, linear independence and eigenvalues/vectors.

For each of these concepts I consider four aspects for analysis,

i. A mathematical account of the concept

ii. The lecturer’s mathematical treatment of the concept in the course notes

iii. The lecturer’s didactical thinking in relation to the teaching of the concept

iv. The lecturer’s teaching of each concept (to students)

I describe each of the four aspects and what they entailed for my analysis in more detail in the next section.