Linear Algebra

Section I. Isomorphisms 161
1.3 Definition An isomorphism between two vector spaces V and W is a map
f : V → W that
(1) is a correspondence: f is one-to-one and onto; ∗
(2) preserves structure: if �v1,�v2 ∈ V then
and if �v ∈ V and r ∈ R then
f(�v1 + �v2) =f(�v1)+f(�v2)
f(r�v) =rf(�v)
(we write V ∼ = W ,read“V is isomorphic to W ”, when such a map exists).
(“Morphism” means map, so “isomorphism” means a map expressing sameness.)
1.4 Example The vector space G = {c1 cos θ + c2 sin θ � � c1,c2 ∈ R} of functions
of θ is isomorphic to the vector space R2 under this map.
c1 cos θ + c2 sin θ f
� �
c1
↦−→
We will check this by going through the conditions in the definition.
We will first verify condition (1), that the map is a correspondence between
the sets underlying the spaces.
To establish that f is one-to-one, we must prove that f(�a) =f( � b) only when
�a = � b.If
then, by the definition of f,
f(a1 cos θ + a2 sin θ) =f(b1 cos θ + b2 sin θ)
� � � �
a1 b1
=
a2
from which we can conclude that a1 = b1 and a2 = b2 because column vectors are
equal only when they have equal components. We’ve proved that f(�a) =f( �b) implies that �a = �b, which shows that f is one-to-one.
To check that f is onto we must check that any member of the codomain R2 mapped to. But that’s clear—any
� �
x
∈ R
y
2
is the image, under f, of this member of the domain: x cos θ + y sin θ ∈ G.
Next we will verify condition (2), that f preserves structure.
∗ More information on one-to-one and onto maps is in the appendix.
b2
c2