Context: In LA6.26 several vector spaces are given. The question is about finding their dimensions. Student Andrew suggests finding a basis of the vector space and counting the number of vectors in it: this number is the dimension of the vector space. The tutor agrees with him.

Student Jack found out correctly all the dimensions, except one, but what the tutor says he wishes to discuss here is their writing. Andrew's presentation was 'a bit insecure', says the tutor, and he used more complicated notation than was needed. He then reminds the students of a set of matrices he mentioned earlier in the term which serves as good notation for LA6.26:

Jack then offers E_ii, i=1,... n, as a basis for the set of diagonal matrices. The tutor says he agrees and asks for a basis for the set of symmetric matrices. Andrew protests that he is 'not quite sure what happens' and asks 'What does the definition of E_pq mean?'. Jack explains 'it's all zero except the pq-th entry' and the tutor apologetically rewrites x_ij as

He also draws E_pq on the b/b as a matrix with all its entries equal to zero except the pq-th entry which is 1. For the symmetric matrices Jack starts manufacturing a basis by including E_ii. Andrew suggests 'taking the sum of E_pq' but the tutor does not look pleased. Andrew explains that he wants to 'exploit the zeros and ones in Epq' in order to 'write down all matrices' but Jack notes this will not happen by 'adding them'. The tutor reminds them of the requirements for a basis (span and linear independence) and Jack suggests E_ii and E_ij+E_ji for i<j, for i and j from 1 to n. These are n+n(n-1)/2= n(n+1)/2. For the antisymmetric matrices Jack suggests the same set of matrices except the E_ii, hence the dimension is n(n-1)/2. His suggestions are accepted.

The set of matrices with Trace zero seems to be the most problematic of all (see fig.4 for the solution presented in the tutorial): Jack suggests Eij such that i¹j and he stops in order to think further. Andrew suggests taking these 'together with E_ii from 1 to n-1'. The tutor disagrees: what is Tr(E_ii)? Andrew replies it is zero but with the tutor's reminding of what the trace is he changes his mind to Tr(E_ii)=1. A few silent seconds follow. Then Jack whispers: 'It's one less because... ' but Andrew is still struggling with Tr(E_ii)=1.

A: Why is it one?

T: Because you told me E_ii.

A: Yes, but I was thinking of E_ii from 1 to n-1.

Jack repeats the definition of E_ij and the tutor stresses that E_ii do not belong to the vector space of matrices with trace zero. To their silence about how the diagonal elements of the matrices with trace zero will be expressed, the tutor suggests E_ii-E_nn where i=1,..., n-1 are matrices with trace zero. These with the matrices Eij for i¹ j will form a basis. Span and linear independence are obvious but to Andrew's request the tutor presents them on the b/b. The tutor concludes by outlining the method for finding a basis: 'the principle is to exhibit a set of elements of the set that span the set and are linearly independent'.