Page 262 - Advanced engineering mathematics

242 CHAPTER 7 Matrices and Linear Systems

from R to R is not linear. Generally a function is nonlinear (fails to be linear) when it involves
2
4
products or powers of the coordinates, or nonlinear functions such as trigonometric functions and
exponential functions, whose graphs are not straight lines.
We will use the notation
n
T : R → R m
m
n
to indicate that T is a linear transformation from R to R .
n
n
m
Every linear transformation T : R → R must map the zero vector O n of R to the zero
m
vector O m of R . To see why this is true, use the linearity of T to write
T (O n ) = T (O n + O n ) = T (O n ) + T (O n ),
so
T (O n ) = O m .
However, a linear transformation may take nonzero vectors to the zero vector. For example, the
linear transformation

T (x, y) = (x − y,0)
2
2
from R to R maps every vector < x, x > to < 0,0 >.
m
n
We will define two important properties that a linear transformation T : R → R may
exhibit.
n
m
T is onto if every vector in R is the image of some vector in R under T . This means that,
m
n
if v is in R , then there must be some u in R such that T (u) = v.
T is one-to-one,or1 − 1, if the only way T (u 1 ) can equal T (u 2 ) is for u 1 = u 2 .This
n
means that two vectors in R cannot be mapped to the same vector in R by T .
m
The notions of one-to-one and onto are independent. A linear transformation may be one-
to-one and onto, one-to-one and not onto, onto and not one-to-one, or neither one-to-one or
onto.

EXAMPLE 7.34
Let
T (x, y) =< x − y,0,0 >.
3
2
Then T is a linear transformation from R to R . T is certainly not one-to-one, since, for example,
T (1,1) = T (2,2) =< 0,0,0 >.
3
In fact, T (x, x) =< 0,0,0 > for every number x. Thus T maps many vectors to the origin in R .
3
3
T is also not onto R , since no vector in R with a nonzero second or third component is the
image of any vector in R under T .
2
EXAMPLE 7.35
3
2
Let S : R → R be defined by
S(x, y, z) =< x, y >.

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October 14, 2010 14:23 THM/NEIL Page-242 27410_07_ch07_p187-246