Real number field axioms (possibly excluding the least upper bound axiom); Vectors in two- and three-dimensional real space; R3 as a vector space over 
; Introduction to real vector spaces and subspaces; Planes and lines; Notion of linear dependence and independence, especially as seen in R2 and R3; Spanning sets, linear transformations and their matrices; Change of bases; linear equations and matrices; row echelon form and its applications to linear equations, rank and inverse; Determinants.