ABSTRACT
At this point we return to the geometric roots of linear algebra; throughout this section we will consider vectors in the original geometric sense as equivalence classes of arrows and denote by V the space of all these vectors. In the introductory section on affine geometry we studied intersections of lines and planes and saw how problems of affine geometry could be translated into algebraic ones by fixing a basis and expressing vectors in terms of their coordinates with respect to this basis. Now in geometry we want to do more than just determine the intersections of fines and planes. In particular, we are interested in metrical quantities like lengths, areas, volumes and angles. Again, to describe such quantities we want to pursue the approach of introducing a basis and then translate geometrical statements into algebraic ones. It turns out that for the discussion of metrical properties a special type of coordinate systems is particularly useful.
