A First Course in Linear Algebra, 2017a

288 Linear Transformations
Definition 5.29: One to One
Suppose ⃗x 1 and ⃗x 2 are vectors in R n . A linear transformation T : R n ↦→ R m is called one to one
(often written as 1 − 1) if whenever⃗x 1 ≠⃗x 2 it follows that :
T (⃗x 1 ) ≠ T (⃗x 2 )
Equivalently, if T (⃗x 1 )=T (⃗x 2 ), then ⃗x 1 =⃗x 2 . Thus, T is one to one if it never takes two different
vectors to the same vector.
The second important characterization is called onto.
Definition 5.30: Onto
Let T : R n ↦→ R m be a linear transformation. Then T is called onto if whenever⃗x 2 ∈ R m there exists
⃗x 1 ∈ R n such that T (⃗x 1 )=⃗x 2 .
We often call a linear transformation which is one-to-one an injection. Similarly, a linear transformation
which is onto is often called a surjection.
The following proposition is an important result.
Proposition 5.31: One to One
Let T : R n ↦→ R m be a linear transformation. Then T is one to one if and only if T (⃗x)=⃗0 implies
⃗x =⃗0.
Proof. We need to prove two things here. First, we will prove that if T is one to one, then T (⃗x)=⃗0 implies
that ⃗x =⃗0. Second, we will show that if T (⃗x)=⃗0 implies that ⃗x =⃗0, then it follows that T is one to one.
Recall that a linear transformation has the property that T (⃗0)=⃗0.
Suppose first that T is one to one and consider T (⃗0).
( )
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. If T (⃗x)=⃗0 itmustbe
thecasethat⃗x =⃗0 because it was just shown that T (⃗0)=⃗0 andT is assumed to be one to one.
Now assume that if T (⃗x)=⃗0, then it follows that⃗x =⃗0. If T (⃗v)=T (⃗u), then
T (⃗v) − T(⃗u)=T (⃗v −⃗u)=⃗0
which shows that⃗v −⃗u = 0. In other words,⃗v =⃗u, andT is one to one.
♠
Note that this proposition says that if A = [ A 1 ··· A n
]
then A is one to one if and only if whenever
it follows that each scalar c k = 0.
0 =
n
∑ c k A k
k=1