A First Course in Linear Algebra, 2017a

9.7. Isomorphisms 499
9.7 Isomorphisms
Outcomes
A. Apply the concepts of one to one and onto to transformations of vector spaces.
B. Determine if a linear transformation of vector spaces is an isomorphism.
C. Determine if two vector spaces are isomorphic.
9.7.1 One to One and Onto Transformations
Recall the following definitions, given here in terms of vector spaces.
Definition 9.62: One to One
Let V ,W be vector spaces with⃗v 1 ,⃗v 2 vectors in V . Then a linear transformation T : V ↦→ W is called
one to one if whenever⃗v 1 ≠⃗v 2 it follows that
T (⃗v 1 ) ≠ T (⃗v 2 )
Definition 9.63: Onto
Let V ,W be vector spaces. Then a linear transformation T : V ↦→ W is called onto if for all ⃗w ∈ ⃗W
there exists⃗v ∈ V such that T (⃗v)=⃗w.
Recall that every linear transformation T has the property that T (⃗0) =⃗0. This will be necessary to
prove the following useful lemma.
Lemma 9.64: One to One
The assertion that a linear transformation T is one to one is equivalent to saying that if T (⃗v)=⃗0,
then⃗v = 0.
Proof. Suppose first that T is one to one.
( )
T (⃗0)=T ⃗ 0 +⃗0 = T (⃗0)+T (⃗0)
and so, adding the additive inverse of T (⃗0) to both sides, one sees that T (⃗0)=⃗0. Therefore, if T (⃗v)=⃗0,
itmustbethecasethat⃗v =⃗0 because it was just shown that T (⃗0)=⃗0.
Now suppose that if T (⃗v) =⃗0, then ⃗v = 0. If T (⃗v) =T (⃗u), thenT (⃗v) − T (⃗u) =T (⃗v −⃗u) =⃗0 which
shows that⃗v −⃗u = 0orinotherwords,⃗v =⃗u.
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