Section (iv) Constructing Bases For Various Vector Spaces: the Inadequate Targeting of Essential Intuitive Ideas and Induction Into the Notational Language of Advanced Mathematical Thinking

An Interpretive Account: The Analysis

Difficulties with Notation and Inadequate Targeting of Essential Intuitive Ideas. Throughout the session Andrew seems to be in great difficulty with the notation suggested by the tutor. Jack, on the contrary, seems to assimilate the new notation almost instantly. As a result, while Jack is already manufacturing bases by manipulating the matrices E_pq, Andrew is still uncertain about the definition of E_pq, or having difficulty with why Tr(E_ii)=1.

Jack appears to be more understanding of — and more efficient in resolving — Andrew's difficulty than the tutor: he describes what the elements of E_pq are like in ordinary English. Andrew's immediate understanding brings the tutor to the apologetic position of not only rewriting the definition of E_pq in a less convoluted but equally rigorous way but also of drawing E_pq on the b/b.

So far in the Episode, my impression of Andrew's unease was emphasised by its contrast with Jack's swift performance. Andrew's difficulty can be attributed to the difficulties associated with the notation used in the tutorial. However in his subsequent suggestion 'taking the sum of E_pq' Andrew appears to ignore the more fundamental fact that the elements of a basis of a subspace must be elements of the subspace. This is not true of the 'sum of E_pq' (whatever that means) or the E_ii, none of which are matrices with trace zero.

I think that here Andrew is struggling, not against a complete impasse, but against some very unclear, elusive intuitions. The 'sum of E_pq' is perhaps a hint at the commonly used idea that every nxn matrix E_ii can be written as

ååx_pqE_pq, where p,q=1,...,n

Jack too captures almost successfully (with 'It's one less') the idea that since the n elements of the diagonal do not vary independently — there is one condition connecting them — their degree of independence decreases by one. Finally Andrew's 'Yes, but I was thinking of E_ii from 1 to n-1' seems to be vaguely targeting the idea of E_ii-E_nn but he doesn't accomplish that fully. The tutor eventually delivers a suggestion for a basis of the set of matrices with trace zero. Andrew's continuing discomfort is illustrated in his request for a detailed presentation of how the set proposed by the tutor spans the set of matrices with trace zero and is linearly independent.