Your exposition is very good. If I could make one pedagogical recommendation, it would be that you give a gentle introduction to linear dependence and independe

Your exposition is very good. If I could make one pedagogical recommendation, it would be that you give a gentle introduction to linear dependence and independence. You don't need this to treat the subject in the geometric fashion you're using (it's implicit in talking about calculating lines on the Euclidean plane). But if you explain it well it's a very powerful didactic mechanic for your audience, because then they can generalize the reason why {(1, 0), (-1, 0)} is not a basis of R^2 to arbitrary dimensions n. It also leads you into the neat result that any linearly independent subset S of a vector space V whose cardinality matches the dimension of V is a basis. That being said, I know all too well what happens to good exposition when you try to shove too much into it. So feel free to ignore this; you write well and I think you accomplish the goal of presenting the material in an intuitive way.