Page 261 - Advanced engineering mathematics

7.10 Linear Transformations 241

A function T that maps n-vectors to m-vectors is called a linear transformation if the
following two conditions are satisfied:
1.
T (u + v) = T (u) + T (v)
for all n-vectors u and v, and
2.
T (αu) = αT (u)
for every real number α and all n-vectors u

These two conditions can be rolled into the single requirement that
T (αu + βv) = αT (u) + βT (v)
n
for all real numbers α and β and vectors u and v in R .
A linear transformation is also called a linear mapping.

EXAMPLE 7.33
Define T by
T (x, y) =< x + y, x − y,2x >.
2
3
Then T maps vectors in R to vectors in R . For example,
T (2,−3) =< −1,5,4 > and T (1,1) =< 2,0,2 >.
We will verify that T is a linear transformation. Let
u =< a,b > and v =< c,d >.
Then
u + v =< a + c,b + d >
and
T (u + v) = T (a + c,b + d) =< a + c + b + d,a + c − b − d,2a + 2c >,
while
T (u) + T (v) =< a + b,a − b,2a > + < c + d,c − d,2c >
=< a + b + c + d,a − b + c − d,2a + 2c >
=< a + c + b + d,a + c − b − d,2a + 2c >

= T (a + c,b + d) = T (u + v).
This verifies condition (1) of the definition. For condition (2), let α be any number. Then

T (αu) = T (αa,αb)
=<αa + αb,αa − αb,2αa >
= α< a + b,a − b,2a >= αT (u).

It is easy to check that the function
2
P(a,b,c) =< a ,1,1,sin(a)>

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October 14, 2010 14:23 THM/NEIL Page-241 27410_07_ch07_p187-246