Linear Algebra, 2020a

Section II. Homomorphisms 201
Recall that for any function h: V → W, the set of elements of V that map to
⃗w ∈ W is the inverse image h −1 (⃗w) ={⃗v ∈ V | h(⃗v) =⃗w}. Above, the left side
shows three inverse image sets.
2.5 Example Consider the projection π: R 3 → R 2
⎛ ⎞
x
⎜ ⎟
⎝y⎠
↦−→
π
z
(
)
x
y
which is a homomorphism that is many-to-one. An inverse image set is a vertical
line of vectors in the domain.
R 3 R 2
⃗w
One example is this.
⎛
( )
π −1 1 ⎜
1
⎞
⎟
( )={ ⎝3⎠ | z ∈ R}
3
z
2.6 Example This homomorphism h: R 2 → R 1
( )
x h
↦−→ x + y
y
is also many-to-one. For a fixed w ∈ R 1 the inverse image h −1 (w)
R 2 R 1
w
is the set of plane vectors whose components add to w.