5. One to One Onto Linear Transformations
Section 3.5. One to One Onto Linear Transformations
In this section, we shall record some important properties of one to one and onto linear transformations AV : U . In Section 3.3, it was mentioned that when a linear transformation AV : U is both one to one and onto, the elements of V are in one to one correspondence with the elements of U . One to one onto linear transformations are sometimes called isomorphisms.
The one to one correspondence between V and U means there exists a function fU : V such
that
f Av v for all vV (3.5.1)
The next result we wish to establish is that f is actually a linear transformation. This means that if we are given vv 1 , 2
V and if we label the corresponding elements in U by
u 1 Av 1 and u 2 Av 2 (3.5.2)
then
f u 1 u 2 fu 1 fu 2 (3.5.3)
for all , C and all uu 1 , 2 in U .
The proof of (3.5.3) goes as follows. First, form the left side of (3.5.3) and use the properties of A as a linear transformation. The result is
f u 1 u 2 f Av 1 Av 2
fA v 1 v 2 (3.5.4)
where (3.5.1) has been used. Next, it follows from (3.5.1) and (3.5.2) that
v 1 f Av 1 fu 1 (3.5.5)
and
v 2 f Av 2 fu 2 (3.5.6)
Equations (3.5.5) and (3.5.6), when utilized in (3.5.4) yield the result (3.5.3).
Chap. 3 •
LINEAR TRANSFORMATIONS
As with matrices, it is customary to denote the inverse linear transformation of a one to one
1 onto linear transformation 1 AV : by A . Also, the linear transformation A : U
V is one
to one and onto whose inverse obeys
A A (3.5.7)
Theorem 3.5.1. If AV : U and BU : W are one to one linear transformations, then
BA V : W is a one to one onto linear transformation whose inverse is computed by
BA AB (3.5.8)
Proof. The fact that BA is a one to one and onto follows directly from the corresponding properties of A and B . The fact that the inverse of BA is computed by (3.5.8) follows directly because if
u = Av
and
w = Bu (3.5.9)
then
and -1 v=Au u=Bw (3.5.10)
It follows from (3.5.9) that
w = Bu B Av BAv (3.5.11)
which implies that
v BA w (3.5.12)
It follows from (3.5.10) that
-1 v=Au ABw (3.5.13)
v BA w A B w (3.5.14)
As a result of (3.5.14)
Sec. 3.5 • One to One Onto Linear Transformations
BA AB w=0 for all wW (3.5.15)
which implies (3.5.8).
The notation 1 A for the inverse allows (3.5.1) to be written
1 A
Av v for all vV (3.5.16)
The product definition (3.4.17) allows (3.5.16) to be written
1 A Av v for all v V (3.5.17)
Likewise, for all uU , we can write
1 AA u u for all uU (3.5.18)
The identity linear transformation : IV
V was introduced in Example 3.1.1. Recall that it is
defined by
Iv = v (3.5.19)
for all v in .
V Often it is desirable to distinguish the identity linear transformations on different
vector spaces. In these cases we shall denote the identity linear transformation by I V . It follows
from (3.5.17), (3.5.18) and (3.5.19) that
1 1 AA I
and AA I V (3.5.20)
Conversely, if A is a linear transformation from V toU , and if there exists a linear transformation
BU 1 : V such that AB I
U and BA I V , then A is one to one and onto and B=A . The
proof of this assertion is left as an exercise. As with matrices, linear transformations that have inverses, i.e., one to one onto linear transformations, are referred to as nonsingular.
Exercises
3.5.1 Show that
M A , ee j , k M Aee , k , j (3.5.21)
226
Chap. 3 •
LINEAR TRANSFORMATIONS
for a nonsingular linear transformation AV : V .
3.5.2 In Example 3.2.2, we were given the linear transformation AV : U defined by
2 3 4 1 2 3 Av 4
2 3 b 1 4 b 2
2 b 3 2 2 b 4
1 2 3 4 1 2 3 4 (3.5.22)
1 2 3 for all 4 v e
1 e 2 e 3 e 4 V . Show that A is nonsingular and that the inverse
V is defined by
19 10
Au 7 u 1 u 2 u 3 4 u 4 e 1
u 1 u 2 u 4 e 2
2 u 1 2 u 2 u 3 u 4 e 3
(3.5.23)
1 2 3 for all 4 u u b
1 u b 2 u b 3 u b 4 U.
Sec. 3.6 • Change of Basis for Linear Transformations