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.