Linear Algebra

176 Chapter 3. Maps Between Spaces
3.II Homomorphisms
The definition of isomorphism has two conditions. In this section we will consider
the second one, that the map must preserve the algebraic structure of the
space. We will focus on this condition by studying maps that are required only
to preserve structure; that is, maps that are not required to be correspondences.
Experience shows that this kind of map is tremendously useful in the study
of vector spaces. For one thing, as we shall see in the second subsection below,
while isomorphisms describe how spaces are the same, these maps describe how
spaces can be thought of as alike.
3.II.1 Definition
1.1 Definition A function between vector spaces h: V → W that preserves
the operations of addition
and scalar multiplication
is a homomorphism or linear map.
if �v1,�v2 ∈ V then h(�v1 + �v2) =h(�v1)+h(�v2)
if �v ∈ V and r ∈ R then h(r · �v) =r · h(�v)
1.2 Example The projection map π : R 3 → R 2
⎛ ⎞
x
⎝y⎠
z
π
↦−→
is a homomorphism. It preserves addition
⎛
π( ⎝ x1
⎞ ⎛
⎠+ ⎝ x2
⎞ ⎛
⎠) =π( ⎝ x1
⎞
+ x2
⎠) =
y1
z1
y2
z2
y1 + y2
z1 + z2
� �
x
y
� x1 + x2
y1 + y2
⎛
�
= π( ⎝ x1
⎞ ⎛
⎠)+π( ⎝ x2
⎞
⎠)
and it preserves scalar multiplication.
⎛
π(r · ⎝ x1
⎞ ⎛
y1⎠)
=π( ⎝
z1
rx1
⎞
⎛
� �
rx1
ry1⎠)
= = r · π( ⎝
ry1
rz1
x1
⎞
y1⎠)
z1
Note that this map is not an isomorphism, since it is not one-to-one. For
instance, both �0 and�e3 in R 3 are mapped to the zero vector in R 2 .
1.3 Example The domain and codomain can be other than spaces of column
vectors. Both of these maps are homomorphisms.
y1
z1
y2
z2