9-21 NONSINGULAR LINEAR TRANSFORMATIONS 709
of poalynomials of degree at most h, and find the kernel and range of each
of the transformations 1)D, 1 Dh. + l
() Verify that each of the identities:: ( + 3+ 2)' = (I + 1)(D + 2)f,
ii) (D1) + 1)f- i(Dl) + 1)(D2 - D + 1)f
is true for/ x' = x1, where k is a positive integer.
9. (a) Let U, V he finite dimensaional vector spaces and let T be a linear mapping
of into V. Prove: If dim = dim V and T is onto then T is one-to-one.
If dinI ( = dim V and is one-to-one, then T is onto. [Iint. Use Theoremn
224]
h() Let I he a linear transformation on a finite dimensional space U. Prove: if
I is one-to one, then T is onto. If T is onto, then F is one-to-one,
(c Show that the conclusions in part a) are false if dim U- din 1 = 0.
d) Show that the conclusions in part (a) are false if dim U 1 dim V.
(e) Show that the conclusions in part (b) are false if dim = oc.
10. Let S and 1T he linear transformiations on a vector space t such that ST = 'S.
Prove the following:
a) Kernel + Kernel erl T} C Kernel SF
(b) If Kernel S Kernel 1 is the zero space V, and null S is finite, then T maps
Kernel S one to-one onto itself.
(c) Under the assumptions of part (), {Kernel S + Kernel }F) = Kernel ST.
(d) If U is infinite-dimensional, with basis {uo, uI,.. u,. and S(uo) u,
S(u i)} = u2,, (u2I) = 0 for i = 1, 2... and T(u) = u, T(u2 ) =-- 0,
T(u = u2142 for i = 1, 2.., then ST = TS and Kernel ST 7 {Kernel S
+ Kernel F }. Thus in part (c) the assumption that null S (1or, alternatively,
null F) be finite, is necessary to obtain the conclusion.
(e Show that on (W, Kernel D f {Kernel D) + Kernel D} = Kernel D. Hence
for part (c) the first assumption of part (h) is necessary.
9-21 NONSINGULAR LINEAR TRANSFORMATIONS
A linear transformation T on a vector space U is said to be nonsingular
if T is one-to-one and onto. The identity I and the mappings cI (c # 0), called
scalar mappings, are nonsingular linear transformations. Linear transformations which are not nonsingular are called singular linear transformations.
EXAMPLE 1. (a) The mapping T(x, y) =(x +, y) is a nonsingular
linear transformation on 2, since T(x, y) = (0, 0) if, and only if, x = 9 = 0
(and, hence, TI is one-to-one) and T(a - b, b) = (a, b) (and, hence, T is onto).
(b) The mapping T(f)- erf(x) is a nonsingular linear transformation on
e- 0, 00), since ezf(x) -- 0 if, and only if, f(x) 0, so that Kernel T is the
zero fimetion and T is one-to-one; also T(e -f) = f, so that T is onto.
