Page 255 - Matrix Analysis & Applied Linear Algebra

250 Chapter 4 Vector Spaces

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(c) When V = , determine [I] BB for
           
1 0 0 1 1 1
   
1
,
,
0
B =       , B =       .
1
1
,
,
0

0
0 0 1 0 0 1
   
3
3
4.7.17. Let T : → be the linear operator defined by
T(x, y, z)=(2x − y, −x +2y − z, z − y).
(a) Determine T −1 (x, y, z).
3
(b) Determine [T −1 ] S , where S is the standard basis for .
4.7.18. Let T be a linear operator on an n -dimensional space V. Show that
the following statements are equivalent.
(1) T −1 exists.
(2) T is a one-to-one mapping (i.e., T(x)= T(y) =⇒ x = y ).
(3) N (T)= {0}.
(4) T is an onto mapping (i.e., for each v ∈V, there is an x ∈V
such that T(x)= v ).
Hint: Show that (1) =⇒ (2) =⇒ (3) =⇒ (4) =⇒ (2),
and then show (2) and (4) =⇒ (1).
n
4.7.19. Let V be an n -dimensional space with a basis B = {u i } i=1 .
(a) Prove that a set of vectors {x 1 , x 2 ,..., x r }⊆V is linearly
independent if and only if the set of coordinate vectors
( )
n×1
[x 1 ] B , [x 2 ] B ,. . . , [x r ] B ⊆
is a linearly independent set.
(b) If T is a linear operator on V, then the range of T is the set

R (T)= {T(x) | x ∈V}.

Suppose that the basic columns of [T] B occur in positions

b 1 ,b 2 ,...,b r . Explain why T(u b 1 ), T(u b 2 ),..., T(u b r ) is a
basis for R (T).