Linear Algebra, 2020a

Section II. Homomorphisms 193
So one way to think of ‘homomorphism’ is that we are generalizing ‘isomorphism’
(by dropping the condition that the map is a correspondence), motivated
by the observation that many of the properties of isomorphisms have only to
do with the map’s structure-preservation property. The next two results are
examples of this motivation. In the prior section we saw a proof for each that
only uses preservation of addition and preservation of scalar multiplication, and
therefore applies to homomorphisms.
1.6 Lemma A linear map sends the zero vector to the zero vector.
1.7 Lemma The following are equivalent for any map f: V → W between vector
spaces.
(1) f is a homomorphism
(2) f(c 1 ·⃗v 1 + c 2 ·⃗v 2 )=c 1 · f(⃗v 1 )+c 2 · f(⃗v 2 ) for any c 1 ,c 2 ∈ R and ⃗v 1 ,⃗v 2 ∈ V
(3) f(c 1 ·⃗v 1 + ···+ c n ·⃗v n )=c 1 · f(⃗v 1 )+···+ c n · f(⃗v n ) for any c 1 ,...,c n ∈ R
and ⃗v 1 ,...,⃗v n ∈ V
1.8 Example The function f: R 2 → R 4 given by
⎛ ⎞
( ) x/2
x
f
0
↦−→ ⎜ ⎟
y ⎝x + y⎠
3y
is linear since it satisfies item (2).
⎛
⎞ ⎛ ⎞ ⎛ ⎞
r 1 (x 1 /2)+r 2 (x 2 /2)
x 1 /2
x 2 /2
0
⎜
⎟
⎝r 1 (x 1 + y 1 )+r 2 (x 2 + y 2 ) ⎠ = r 0
1 ⎜ ⎟
⎝x 1 + y 1 ⎠ + r 0
2 ⎜ ⎟
⎝x 2 + y 2 ⎠
r 1 (3y 1 )+r 2 (3y 2 )
3y 1
3y 2
However, some things that hold for isomorphisms fail to hold for homomorphisms.
One example is in the proof of Lemma I.2.4, which shows that
an isomorphism between spaces gives a correspondence between their bases.
Homomorphisms do not give any such correspondence; Example 1.2 shows this
and another example is the zero map between two nontrivial spaces. Instead,
for homomorphisms we have a weaker but still very useful result.
1.9 Theorem A homomorphism is determined by its action on a basis: if V is a vector
space with basis 〈⃗β 1 ,...,⃗β n 〉,ifW is a vector space, and if ⃗w 1 ,...,⃗w n ∈ W
(these codomain elements need not be distinct) then there exists a homomorphism
from V to W sending each ⃗β i to ⃗w i , and that homomorphism is unique.