9-11 LINEAR MAPPINGS 685
T are the same mapping. Thus, in describing a linear mapping, it is only
ne(essary to prescribe its valuest at a basis for the domain space.
EXAMPLE 6. Let a1,...,, a, be real numbers, then the mapping
T( Xl,...,xn) = alx + "'" + a,, X,,
is a linear maopping of V into V1 = R. Conversely, every linear mapping of
Sinto V1 is of this fojr.
PROOF. It is an easy matter to verify that T is indeed a linear mapping
of 1ln into I', and we leave it to the reader to do so.
Let S he a linear mapping of )V, into V1 = R. Suppose that S(el) = bl,...
S(e) b. Then
S(x1,.., x,) = S(x1e 1 + " + x1ne)
= xlS(e1) + + xrS(en) = bixI + + b)nx,
Thus the mapping S has the form asserted.
In Example 6, 1 is th zero mapping if, and only if, all a; = 0. If some
a 0, then T has range R, since T(ca-lei) = c, for each real c.
EXAMPLE 7. Let a1, a, a, b1, b2, b be real numbers. Then the mapping
T(x1, x2, x3) = (alx1 + a2x2 + aax3, bl1x + b2x2 + b3x3)
is a linear mapping of V3 into V2. Conversely, ecvery linear mapping of V
into V,2 is of thfis form.
PROOF. The reader should verify that T is a linear mapping and that the
mapping T is the zero mapping exactly when a1 = a t = a3 = b1 = b2 =
b = 0. ILet S be a linear mapping of V' into V2. Suppose that
5(1, 0, 0) = (C1, dl), (0, 1, 0) = (c2, d2), S(0, 0, 1) = (c3, d3)
Then S has the asserted form, since
S(x, x, x) = x,S(, 0, o) + xS0, 1, 0) + xS(0,0, 1)
= x(c1, d1) + x2e2, d2) + x3(c3, d3)
- (Cx1 + C2x2.,+ c3x3,, dlx1 t d2x2 + d3x3)
EXAMPLE 8. Let T hbe a linear mapping of the space of real polynomials
(P into itself, where T(1) = 0, T(x) = x ' for m > 1. If a(x) =
ao + alx +... + ax, then
T(a(x)) = T(a) + Tr(a1x) +... + T(ax) = a0T(1) +... + aT(x)
= a1 + a, x +... + anx () - a()
x
The range of I is ( since T(xa(x)) = a(x). The mapping T is not
one-to-one, since T(a + a(x)) = T(a(x)) for all real numbers a. The mapping
T is different from the derivative, since T(x2) = x ~ 2x = D(x2).
Notation. For a general mapping T, we denote the set of all values of T(x)
for x in E by T(E).
