Linear Algebra

Section I. Isomorphisms 169
(d) Suppose that U and W are subspaces of a vector space V such that V =
U ⊕ W . Show that the map f : U × W → V given by
f
(�u, �w) ↦−→ �u + �w
is an isomorphism. Thus if the internal direct sum is defined then the internal
and external direct sums are isomorphic.
3.I.2 Dimension Characterizes Isomorphism
In the prior subsection, after stating the definition of an isomorphism, we
gave some results supporting the intuition that such a map describes spaces
as “the same”. Here we will formalize this intuition. While two spaces that
are isomorphic are not equal, we think of them as almost equal—as equivalent.
In this subsection we shall show that the relationship ‘is isomorphic to’ is an
equivalence relation. ∗
2.1 Theorem Isomorphism is an equivalence relation between vector spaces.
Proof. We must prove that this relation has the three properties of being symmetric,
reflexive, and transitive. For each of the three we will use item (2) of
Lemma 1.9 and show that the map preserves structure by showing that the it
preserves linear combinations of two members of the domain.
To check reflexivity, that any space is isomorphic to itself, consider the identity
map. It is clearly one-to-one and onto. The calculation showing that it
preserves linear combinations is easy.
id(c1 · �v1 + c2 · �v2) =c1�v1 + c2�v2 = c1 · id(�v1)+c2 · id(�v2)
To check symmetry, that if V is isomorphic to W via some map f : V → W
then there is an isomorphism going the other way, consider the inverse map
f −1 : W → V . As stated in the appendix, the inverse of the correspondence
f is also a correspondence, so we need only check that the inverse preserves
linear combinations. Assume that �w1 = f(�v1), i.e., that f −1 ( �w1) =�v1, and also
assume that �w2 = f(�v2).
f −1 (c1 · �w1 + c2 · �w2) =f −1� c1 · f(�v1)+c2 · f(�v2) �
= f −1 ( f � c1�v1 + c2�v2) �
= c1�v1 + c2�v2
= c1 · f −1 ( �w1)+c2 · f −1 ( �w2)
Finally, to check transitivity, that if V is isomorphic to W via some map f
and if W is isomorphic to U via some map g then also V is isomorphic to U,
consider the composition map g ◦ f : V → U. As stated in the appendix, the
∗ More information on equivalence relations is in the appendix.