Linear Algebra

182 Chapter Three. Maps Between SpacesRecall that for any function h: V → W, the set of elements of V that map to⃗w ∈ W is the inverse image h −1 (⃗w) = {⃗v ∈ V ∣ ∣ h(⃗v) = ⃗w}. Above, the left sideshows three inverse image sets.2.5 Example Consider the projection π: R 3 → R 2⎛ ⎞x⎜ ⎟⎝y⎠↦−→πz()xywhich is a homomorphism that is many-to-one. An inverse image set is a verticalline of vectors in the domain.One example is this.R 3 R 2⎛( )π −1 1 ⎜1⎞⎟( ) = { ⎝3⎠ ∣ z ∈ R}3z2.6 Example This homomorphism h: R 2 → R 1( )x h↦−→ x + yyis also many-to-one. For a fixed w ∈ R 1 , the inverse image h −1 (w)⃗wR 2 R 1wis the set of plane vectors whose components add to w.In generalizing from isomorphisms to homomorphisms by dropping the oneto-onecondition, we lose the property that we’ve stated intuitively as thatthe domain is “the same” as the range. We lose that the domain correspondsperfectly to the range. What we retain, as the examples below illustrate, is thata homomorphism describes how the domain is “like” or “analogous to” the range.