ABSTRACT
Along with vector spaces we also studied linear mappings as those mappings between vector spaces which preserve the algebraic structure given by addition and scalar multiplication. Now as we study spaces with an extra structure like a scalar product we will mainly be interested in those linear mappings which preserve not just the vector space operations but also this extra structure. Thus we will now study linear mappings f: V → W between Euclidean spaces which respect the scalar product in one way or the other. Throughout the section we will restrict ourselves to the finitedimensional case. In this case, the correspondences between V and V* and between W and W* given by Riesz’ representation theorem allow one to set up a correspondence of Hom(W, V) with Hom(W*, V*) and thus with Hom(V, W).
