I'm currently struggling to teach a 2nd course on linear algebra (in the UK, not at an Oxbridge quality university: the students have done a 1st course which concentrated upon algorithms you can apply to matrices, and calculations involving R^2, R^3 etc.)
The syllabus suggests a very abstract approach, stating and proving the Steinitz exchange lemma, and then working up to showing that every (finite dimensional) vector space has a basis of a fixed size etc. etc.
However, I think I'm killing my students. This is all very abstract, it's going to take me weeks to do, and the end result is: All finite dimensional vector spaces look, well, exactly as you think they do. I'm tempted to skip on to linear maps, matrices etc. which seems more interesting to me (and sort of motivates why we might care about choosing a different basis...)
However, I'm also loathed to just assert these facts without proof: the students saw that before in the previous course, and in a pure maths course, I sort of want to prove things (even if I don't expect the students to understand everything).
What do people think about doing a linear algebra course in maximal abstraction early on? Is there a good book which takes a very streamlined (if perhaps hard to understand) approach to proving the existence of bases-- at least then I could get it over with quickly without lying to the students.
I read http://mathoverflow.net/questions/9402/pedagogical-question-about-linear-algebra which was very useful, but maybe concentrated upon a different problem.