Mathematics 2210 Example: Is Tone-to-one or onto?
Consider the function Twith the rule Tļ£«
ļ£¬
ļ£
ļ£®
ļ£Æ
ļ£°
x1
x2
x3
ļ£¹
ļ£ŗ
ļ£»
ļ£¶
ļ£·
ļ£ø=ļ£®
ļ£Æ
ļ£°
0
x1
x2
ļ£¹
ļ£ŗ
ļ£».
Is Tone-to-one? Is the range of Tall of R3? (That is, is Tonto?) If not,
what is the range of T?
We can answer these questions directly or we can try to apply the theory
from Section 1.9.
It is clear that there are vectors other than the zero vector at which T
has the value 0; e.g., Tļ£«
ļ£¬
ļ£
ļ£®
ļ£Æ
ļ£°
0
0
x3
ļ£¹
ļ£ŗ
ļ£»
ļ£¶
ļ£·
ļ£ø=0no matter what x3is. So Tis not
one-to-one.
Likewise, it is clear that all of the values of Thave first coordinate equal
to zero, so not every vector in R3belongs to the range of T. Thus Tisnāt
onto. To find the exact range of a function is often a difficult task, but in
this example it isnāt hard. If bā R3is in the range of T, then b1= 0. But
conversely, if b1= 0, then we can solve the equation T(x) = b, which is the
system of equations 0 = b1, x1=b2, x2=b3and see that there are solutions.
(The general solution is x= (b2, b3, t) with tarbitrary, i.e., any real number.)
So every such bis a member of the range. Conclusion: In this example, the
range of the transformation is the set of all bā R3for which b1= 0; i.e.,
Range(T) = Span{e2, e3}.
Since the matrix of Tis
A=ļ£®
ļ£Æ
ļ£°
0 0 0
1 0 0
0 1 0
ļ£¹
ļ£ŗ
ļ£»
(Explain why this is so.), we could note that the columns of Aare linearly
dependent (since one of them is a zero vector) and not every row of Acontains
a pivot position. Hence by theorems in Sec. 1.9, Tisnāt one-to-one or onto.