Quotient transformations

Suppose that is a linear transformation on a vector space and that is a subspace of invariant under . Under these circumstances there is a natural way of defining a linear transformation (to be denoted by ) on the space ; this "quotient transformation" is related to just about the same way as the quotient space is related to . It will be convenient (in this section) to denote by the more compact symbol , and to use related symbols for the vectors and the linear transformations that occur. Thus, for instance, if is any vector in , we shall denote the coset by ; objects such as are the typical elements of .

To define the quotient transformation (to be denoted, alternatively, by ), write for every vector in . In other words, to find the transform by of the coset , first find the transform by of the vector , and then form the coset of determined by that transformed vector. This definition must be supported by an unambiguity argument; we must be sure that if two vectors determine the same coset, then the same is true of their transforms by . The key fact here is the invariance of . Indeed, if , then is in , so that (invariance) is in , and therefore .

What happens if is not merely invariant under , but, together with a suitable subspace , reduces ? If this happens, then is the direct sum, say , of two linear transformations defined on the subspaces and of , respectively; the question is, what is the relation between and ? Both these transformations can be considered as complementary to ; the transformation describes what does on , and both and describe in different ways what does elsewhere.

Let be the correspondence that assigns to each vector in the coset ( ). We know already that is an isomorphism between and (cf. Section: Dimension of a quotient space , Theorem 1); we shall show now that the isomorphism carries the transformation over to the transformation . If (where, of course, is in ), then it follows that This implies that , as promised. Loosely speaking (see Section: Similarity ) we may say that transforms the same way as transforms . In other words, the linear transformations and are abstractly identical (isomorphic). This fact is of great significance in the applications of the concept of quotient space.