Quotient transformations

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

To define the quotient transformation $A/ℳ$ (to be denoted, alternatively, by $A^{-}$ ), write $$A^{-}{x}^{-}=(Ax{)}^{-}$$ for every vector $x$ in $𝒱$ . In other words, to find the transform by $A/ℳ$ of the coset $x+ℳ$ , first find the transform by $A$ of the vector $x$ , and then form the coset of $ℳ$ 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 $ℳ$ . Indeed, if $x+ℳ=y+ℳ$ , then $x-y$ is in $ℳ$ , so that (invariance) $Ax-Ay$ is in $ℳ$ , and therefore $Ax+ℳ=Ay+ℳ$ .

What happens if $ℳ$ is not merely invariant under $A$ , but, together with a suitable subspace $𝒩$ , reduces $A$ ? If this happens, then $A$ is the direct sum, say $A=B\oplus C$ , of two linear transformations defined on the subspaces $ℳ$ and $𝒩$ of $𝒱$ , 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 $ℳ$ , 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 $𝒩$ the coset $x^{-}$ ( $=x+ℳ$ ). We know already that $T$ is an isomorphism between $𝒩$ and $𝒱/ℳ$ (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 $Cx=y$ (where, of course, $x$ is in $𝒩$ ), then $$A^{-}{x}^{-}=(Ax{)}^{-}=(Cx{)}^{-}=y^{-};$$ it follows that $$TCx=Ty=A^{-}Tx.$$ This implies that $TC=A^{-}T$ , as promised. Loosely speaking (see Section: Similarity ) we may say that $A^{-}$ transforms ${𝒱}^{-}$ the same way as $C$ transforms $𝒩$ . 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.