Linear Algebra

178 Chapter 3. Maps Between Spaces
Obviously, any isomorphism is a homomorphism—an isomorphism is a homomorphism
that is also a correspondence. So, one way to think of the ‘homomorphism’
idea is that it is a generalization of ‘isomorphism’, motivated by the
observation that many of the properties of isomorphisms have only to do with
the map respecting structure and not to do with it being a correspondence. As
examples, these two results from the prior section do not use one-to-one-ness or
onto-ness in their proof, and therefore apply to any homomorphism.
1.6 Lemma A homomorphism sends a zero vector to a zero vector.
1.7 Lemma Each of these is a necessary and sufficient condition for f : V → W
to be a homomorphism.
(1) for any c1,c2 ∈ R and �v1,�v2 ∈ V ,
f(c1 · �v1 + c2 · �v2) =c1 · f(�v1)+c2 · f(�v2)
(2) for any c1,...,cn ∈ R and �v1,... ,�vn ∈ V ,
f(c1 · �v1 + ···+ cn · �vn) =c1 · f(�v1)+···+ cn · f(�vn)
This lemma simplifies the check that a function is linear since we can combine
the check that addition is preserved with the one that scalar multiplication is
preserved and since we need only check that combinations of two vectors are
preserved.
1.8 Example The map f : R 2 → R 4 given by
� �
x f
↦−→
y
⎛ ⎞
x/2
⎜ 0 ⎟
⎝x
+ y⎠
3y
satisfies that check
⎛
⎞ ⎛
r1(x1/2) + r2(x2/2)
⎜
0
⎟ ⎜
⎟
⎝r1(x1
+ y1)+r2(x2 + y2) ⎠ = r1
⎜
⎝
r1(3y1)+r2(3y2)
and so it is a homomorphism.
x1/2
0
x1 + y1
3y1
⎞ ⎛
⎟ ⎜
⎟
⎠ + r2
⎜
⎝
x2/2
0
x2 + y2
(Sometimes, such as with Lemma 1.15 below, it is less awkward to check preservation
of addition and preservation of scalar multiplication separately, but this
is purely a matter of taste.)
However, some of the results that we have seen for isomorphisms fail to hold
for homomorphisms in general. An isomorphism between spaces gives a correspondence
between their bases, but a homomorphisms need not; Example 1.2
shows this and another example is the zero map between any two nontrivial
spaces. Instead, a weaker but still very useful result holds.
3y2
⎞
⎟
⎠