Suppose that A is a linear transformation on a vector space \mathcal{V} and that \mathcal{M} is a subspace of \mathcal{V} invariant under A . Under these circumstances there is a natural way of defining a linear transformation (to be denoted by A / \mathcal{M} ) on the space \mathcal{V} / \mathcal{M} ; this "quotient transformation" is related to A just about the same way as the quotient space is related to \mathcal{V} . It will be convenient (in this section) to denote \mathcal{V} / \mathcal{M} by the more compact symbol \mathcal{V}^{-} , and to use related symbols for the vectors and the linear transformations that occur. Thus, for instance, if x is any vector in \mathcal{V} , we shall denote the coset x+\mathcal{M} by x^{-} ; objects such as x^{-} are the typical elements of \mathcal{V}^{-} .
To define the quotient transformation A / \mathcal{M} (to be denoted, alternatively, by A^{-} ), write A^{-} x^{-}=(A x)^{-} for every vector x in \mathcal{V} . In other words, to find the transform by A / \mathcal{M} of the coset x+\mathcal{M} , first find the transform by A of the vector x , and then form the coset of \mathcal{M} 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 A . The key fact here is the invariance of \mathcal{M} . Indeed, if x+\mathcal{M}=y+\mathcal{M} , then x-y is in \mathcal{M} , so that (invariance) A x-A y is in \mathcal{M} , and therefore A x+\mathcal{M}=A y+\mathcal{M} .
What happens if \mathcal{M} is not merely invariant under A , but, together with a suitable subspace \mathcal{N} , reduces A ? If this happens, then A is the direct sum, say A=B \oplus C , of two linear transformations defined on the subspaces \mathcal{M} and \mathcal{N} of \mathcal{V} , respectively; the question is, what is the relation between A^{-} and C ? Both these transformations can be considered as complementary to A ; the transformation B describes what A does on \mathcal{M} , and both A^{-} and C describe in different ways what A does elsewhere.
Let T be the correspondence that assigns to each vector x in \mathcal{N} the coset x^{-} ( =x+\mathcal{M} ). We know already that T is an isomorphism between \mathcal{N} and \mathcal{V} / \mathcal{M} (cf. Section: Dimension of a quotient space , Theorem 1); we shall show now that the isomorphism carries the transformation C over to the transformation A^{-} . If C x=y (where, of course, x is in \mathcal{N} ), then A^{-} x^{-}=(A x)^{-}=(C x)^{-}=y^{-}; it follows that T C x=T y=A^{-} T x. This implies that T C=A^{-} T , as promised. Loosely speaking (see Section: Similarity ) we may say that A^{-} transforms \mathcal{V}^{-} the same way as C transforms \mathcal{N} . In other words, the linear transformations A^{-} and C are abstractly identical (isomorphic). This fact is of great significance in the applications of the concept of quotient space.
